An $L(2,1)$-labelling of a finite graph $Γ$ is a function that assigns integer values to the vertices $V(Γ)$ of $Γ$ (colouring of $V(Γ)$ by ${\mathbb{Z}}$) 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 $λ(Γ)$ of $Γ$ measures the least number of integers needed for such a labelling (colouring). A power graph $Γ_G$ 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 $λ(Γ_G) \geq |G|$ for any finite group. In this paper we show that if $G$ is a finite group of a prime power order, then $λ(Γ_G) = |G|$ 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.