An L(2, 1)-labelling of a finite graph Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} is a function that assigns integer values to the vertices V(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$V(\varGamma )$$\end{document} of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} (colouring of V(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$V(\varGamma )$$\end{document} by Z\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}$$\end{document}) so that the absolute difference of two such values is at least 2 for adjacent vertices and is at least 1 for vertices, which are precisely distance 2 apart. The lambda number λ(Γ)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (\varGamma )$$\end{document} of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma $$\end{document} measures the least number of integers needed for such a labelling (colouring). A power graph ΓG\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma _G$$\end{document} of a finite group G is a graph with vertex set as the elements of G and two vertices are joined by an edge if and only if one of them is a positive integer power of the other. It is known that λ(ΓG)≥|G|\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (\varGamma _G) \ge |G|$$\end{document} for any finite group. In this paper, we show that if G is a finite group of a prime power order, then λ(ΓG)=|G|\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (\varGamma _G) = |G|$$\end{document} if and only if G is neither cyclic nor a generalized quaternion 2-group. This settles a partial classification of finite groups achieving the lower bound of lambda number.
