  Let $ G $ be a finite group of order $ n$. The strong power graph
$\mathcal{P}_s(G) $ of $G$ is the undirected graph whose vertices are the
elements of $G$ such that two distinct vertices $a$ and $b$ are adjacent if
$a^{{m}_1}$=$b^{{m}_2}$ for some positive integers ${m}_1 ,{m}_2 < n$. In this
article we classify all groups $G$ for which $\mathcal{P}_s(G)$ is line graph
and Caley graph. Spectrum and permanent of the Laplacian matrix of the strong
power graph $\mathcal{P}_s(G)$ are found for any finite group $G$.
