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