  Let $G$ be a finite $p$-group, and let $\der Gd$ denote the $d$-th term of
the derived series of $G$. We show, for $p\geq 5$, that $\der Gd\neq 1$ implies
$\log_p\ord G\geq 2^d+3d-6$, and hence we improve a recent result by Mann.
