There is some facts about finite non abelian $p$-groups over the site. For example, when $n=3$: Nonabelian groups of order $p^3$. I have found the following problem in my very old works unsolved, claiming:
$G$ is a finite non-abelian $p$-group, $|G|=p^n$. Then $|G'|\neq p^{n-1}$
When $p=3$ we start with $Z(G)\neq 1$ to show that $|Z(G)|=|G'|=p$. But in above problem it seems to me that induction on all $p$-groups (finite and non abelian) with orders less than $n$ may be applicable. In fact for such a group, $|G|=p^n$, we have:$|Z(G)|=p, p^2,... \mathrm{or}\ p^{n-2}$Is my approach valid? Thanks.