Exhibit 5.1

	Lucas GC Limited	Email dbulley@applebyglobal.com
	Palm Grove Unit 4 265 Smith Road, George Town P.O. Box 52A Edgewater Way, #1653 Grand Cayman KY1-9006 Cayman Islands	Tel+852 2523 8123
		Appleby Ref 473424.0001

		10 August 2026

Suites 3504B-06

35/F, Two Taikoo Place

979 King’s Road

Quarry Bay

Hong Kong

Tel +852 2523 8123

applebyglobal.com

Managing Partner

David Bulley

Partners

Fiona Chan

Kitty Chan

Vincent Chan

Chris Cheng

Richard Grasby

Judy Lee

Michael Makridakis

John McCarroll SC

Lorinda Peasland

Eliot Simpson

Lucas GC Limited (Company)

INTRODUCTION

This opinion as to Cayman Islands law is addressed to you in connection with the Company’s filing of prospectus supplement to the registration statement on Form F-3 (File No.: 333-286651) (Registration Statement), filed with the United States Securities and Exchange Commission (Commission) under the United States Securities Act of 1933, as amended (Securities Act) and a Form 6-K relating to an at-the-market offering program (Offering) by the Company to the Agent (as defined below) of the Class A ordinary shares (Offer Shares), par value $0.0002 per share of the Company, having an aggregate offering price of up to $25,000,000 from time to time through or to the Agent, acting as sales agent and/or principal.

OUR REVIEW

For the purposes of giving this opinion we have examined and relied (without further verification) upon the documents listed in Schedule 1 (Documents). We have not examined any other documents, even if they are referred to in the Documents.

We have not made any other enquiries concerning the Company and in particular we have not investigated or verified any matter of fact or opinion (whether set out in any of the Documents or elsewhere) other than as expressly stated in this opinion. Furthermore, we have made no independent investigation of any laws other than the laws of the Cayman Islands.

Unless otherwise defined herein, capitalised terms have the meanings assigned to them in Schedule 1.

		LIMITATIONS

		Our opinion is limited to, and should be construed in accordance with, the laws of the Cayman Islands at the date of this opinion. We express no opinion on the laws of any other jurisdiction.

		This opinion is limited to the matters stated in it and does not extend, and is not to be extended by implication, to any other matters. We express no opinion on the commercial implications of the Documents or whether they give effect to the commercial intentions of the parties. Except as specifically stated herein, we make no comment with respect to any representations and warranties which may be made by or with respect to the Company in any of the documents or instruments cited in this opinion or otherwise with respect to the commercial terms of the transactions the subject of this opinion.

		This opinion is given solely for the benefit of the Company in connection with the matters referred to herein and may be relied upon only by the Company, the Company’s legal advisers in that capacity and purchasers of the Offer Shares pursuant to the Registration Statement. Except with our prior written consent, this opinion may not be used or relied upon by any other person. This opinion is issued solely for the purposes of the filing of the Registration Statement and the Offering by the Company and is not to be relied upon in respect of any other matter.

		This opinion may be used only in connection with the Offering while the Registration Statement is effective.

		We hereby consent to the filing of this opinion as an exhibit to the Form 6-K. In giving this consent we do not hereby admit that we are experts within the meaning of Section 11 of the Securities Act or that we are within the category of persons whose consent is required under Section 7 of the Securities Act or the Rules and Regulations of the Commission promulgated thereunder.

		ASSUMPTIONS AND RESERVATIONS

		We give the following opinions on the basis of the assumptions set out in Schedule 2 (Assumptions), which we have not verified, and subject to the reservations set out in Schedule 3 (Reservations).

		OPINIONS

		1.	Incorporation and Status: The Company is an exempted company incorporated with limited liability and existing under the laws of the Cayman Islands and is a separate legal entity.

		2.	Issue of Offer Shares: The Offer Shares to be allotted and issued by the Company have been duly authorized, and when fully paid, allotted and issued by the Company in the manner set out in the Registration Statement and the Sales Agreement and in accordance with the Resolutions, will be validly issued, fully paid and non-assessable. The reference in this opinion to shares being non-assessable shall mean solely that no further sums of money are required to be paid by the holders of such shares in connection with the issuance thereof.

		3.	Authorisation: The Company has taken all necessary corporate action to authorise the execution and delivery of the Sales Agreement and the performance of the Company’s obligations thereunder.

	Yours faithfully

	/s/ Appleby
	Appleby

Schedule 1

Part 1

Transaction Document

A draft of the at the market offering agreement (the Sales Agreement) between the Company and Pacific Century Securities, LLC (the Agent) provided to us on 7 August 2026.

Part 2

Other Documents Examined

	1.	A copy of the certificate of incorporation of the Company dated 18 August 2022 and a copy of the certificate of incorporation on change of name of the Company dated 27 October 2022, each issued by the Registrar of Companies (together, Certificate of Incorporation).

	2.	A copy of the Certificate of Good Standing dated 25 June 2026 issued by the Registrar of Companies in respect of the Company (Certificate of Good Standing).

	3.	A copy of the amended and restated memorandum of association and articles of association of the Company adopted by special resolution passed on 5 December 2025 (Constitutional Documents).

	4.	PDF copies of the written resolutions of the board of directors of the Company dated 9 April 2026 and the written resolutions of the board of directors of the Company dated 13 July 2026 regarding approval of the Offering (together, Resolutions).

	5.	The Registration Statement.

	6.	A copy of the Register of Directors and Officers of the Company provided to us on 27 April 2026 (Register of Directors and Officers).

Schedule 2

Assumptions

	We have assumed:

	1.	(i) that the originals of all documents examined in connection with this opinion are authentic, accurate and complete; and (ii) the authenticity, accuracy, completeness and conformity to original documents of all documents submitted to us as copies;

	2.	that the signatures, initials and seals on all documents and certificates submitted to us as originals or copies of executed originals are authentic, and the signatures and initials on any Sales Agreement executed by the Company are the signatures and initials of a person or persons authorised by the Company under the Resolutions to execute such Sales Agreement;

	3.	that where incomplete documents, drafts or signature pages only have been supplied to us for the purposes of issuing this opinion, the original documents have been duly completed and correspond in all material respects with the last version of the relevant documents examined by us prior to giving our opinion;

	4.	that the Registration Statement, the Sales Agreement and the Constitutional Documents do not differ in any material respects from any draft of the same which we have examined and upon which this opinion is based;

	5.	that each of the parties to the Sales Agreement (other than the Company under Cayman Islands law) is incorporated, organised or registered (as the case may be) and in good standing (where such concept is legally relevant) under the laws which govern its capacity and has the capacity, power and authority, has fulfilled all internal authorisation procedures and completed all applicable filings and formalities, and has obtained all authorisations, approvals, consents, licences and exemptions required under the laws of any relevant jurisdiction to execute, deliver and perform its respective obligations under Sales Agreement and the transactions contemplated thereby and has taken all necessary corporate and other action required and completed all applicable formalities required to authorise the execution of the Sales Agreement and the performance of its obligations under them;

	6.	the due execution and delivery of the Sales Agreement by each of the parties thereto (other than execution by the Company under Cayman Islands law);

	7.	that the Sales Agreement constitutes, or, when executed, will constitute, legal, valid, binding and enforceable obligations of all parties thereto (save for the Company under Cayman Islands law) in accordance with their governing law;

	8.	that the choice of laws as the governing law in the Sales Agreement has been made in good faith and is valid and binding under the laws of all relevant jurisdictions (other than the Cayman Islands);

	9.	that, insofar as any obligation under the Sales Agreement is to be performed by any of the parties thereto in any jurisdiction outside of the Cayman Islands, its performance will be legal and effective in accordance with the law of any jurisdiction to which it is subject or in which it is constituted and established;

	10.	that no party to the Sales Agreement by having entered into and performing the transactions contemplated by the Sales Agreement will be in breach of any other agreement, deed, trust deed or licence to which it is a party or by which it is bound;

	11.	the truth, accuracy and completeness of all representations and warranties or statements of fact or law (other than as to the laws of the Cayman Islands in respect of matters upon which we have expressly opined) made in the Sales Agreement;

	12.	that the Register of Directors and Officers accurately reflects the names of all directors and officers of the Company as at the dates the Resolutions were passed or adopted, the date the Sales Agreement was executed and as at the date of this opinion;

	13.	that no monies paid to or for the account of any party under the Sales Agreement or any property received or disposed of by any party to the Sales Agreement in each case in connection with the Sales Agreement or the consummation of the transactions contemplated thereby represent or will represent proceeds of criminal conduct or criminal property or terrorist property (as defined in the Proceeds of Crime Act (As Revised) and the Terrorism Act (As Revised), respectively);

	14.	no invitation has been or will be made by or on behalf of the Company to the public in the Cayman Islands to subscribe for any of the Offer Shares;

	15.	the Company will receive money or money’s worth in consideration for the issue of the Offer Shares; and

	16.	that the Registration Statement has been, or will be, declared effective by the Commission prior to the issuance of the Offer Shares.

SCHEDULE 3

Reservations

	Our opinion is subject to the following:

	1.	Title: Legal title to the Offer Shares is determined (in the absence of fraud, manifest error or other extraordinary circumstances) by reference to the register of members maintained by the Company (usually but not always at its registered office); however, there are certain limited circumstances where an application may be made to a Cayman Islands court for a determination on whether the register of members reflects the correct legal position. Further, the Cayman Islands court has the power to order that the register of members maintained by a company should be rectified where it considers that the register of members does not reflect the correct legal position. As far as we are aware, such applications are rarely made in the Cayman Islands and for the purposes of the opinion given in paragraph 2, there are no circumstances or matters of fact known to us on the date of this opinion which would properly form the basis for an application for an order for rectification of the register of members of the Company, but if such an application were made in respect of the Company’s Offer Shares, then the validity of such shares may be subject to re-examination by a Cayman Islands court.

	2.	Non-Assessable: In this opinion the phrase “non-assessable” means, with respect to the issuance of any shares, that a shareholder shall not, in respect of the relevant shares and in the absence of any contractual arrangements, or any obligations pursuant to the memorandum and articles of association, to the contrary, have any obligation to make further contributions to the Company’s assets (except in exceptional circumstances, such as involving fraud, the establishment of an agency relationship or an illegal or improper purpose or other circumstance in which a court may be prepared to pierce or lift the corporate veil).

	3.	Corporate Documents: The Registry of Companies in the Cayman Islands is not public in the sense that copies of the Company’s Constitutional Documents and information on members (shareholders) is not publicly available and information on directors is limited. We have therefore obtained copies of the corporate documents specified in Schedule 1 and relied exclusively on such copies for the verification of such corporate information.

	4.	Issue of Shares: The English case of Houldsworth v City of Glasgow Bank (1880) 5 App Cas 317 HL, provided that (i) in the event of a misrepresentation by a company on which a shareholder relied in agreeing to subscribe for shares in such company, the shareholder may be entitled to rescind the share subscription agreement and thereafter claim damages against such company for any additional loss suffered as a result of the misrepresentation; (ii) such a claim for damages will not arise unless and until the shareholder has successfully rescinded the share subscription agreement; and (iii) that a shareholder may be barred from rescinding on the grounds of delay or affirmation and if such company is wound up (whether voluntarily or compulsorily), such shareholder will lose the right to rescind the share subscription agreement (The Rule of Houldsworth). The Rule of Houldsworth was expressly not followed by the Cayman Islands Grand Court in a first instance decision (currently under appeal). Our assessment is that the Rule of Houldsworth as framed above is of questionable status in the Cayman Islands and if a company enters winding up (whether voluntarily or compulsorily) a shareholder would not necessarily lose the right to rescind the share subscription agreement.