Theorem. Let $G$ be a finite, non-abelian $p$-group all of whose proper subgroups are abelian. Then $|G'|=p$.
Take a counterexample of minimal order. Assume that exist a $H$ such that $1
Then? How am I supposed to continue?
Edit
Additional infos
$G'$ is elementary abelian since $G$ is Frattini-in-center.