How does we obtain a small (not necessarily smallest) value of $n$ such that a $p$-group of order $p^m$ can be embedded in $\operatorname{GL}(n,\mathbb{F}_p)$?
[For embedding of a finite group $G$ in $S_n$, we can compute a small value of $n$ by looking at the minimum index of a subgroup $H$ containing no proper normal subgroup of $G$. Here, this value of $n$ need be necessarily smallest.]