  More than 70 years ago, P. Hall showed that if $G$ is a finite $p$-group such
that a term $\der G{d+1}$ of the derived series is non-trivial, then the order
of the quotient $\der Gd/\der G{d+1}$ is at least $p^{2^d+1}$. Recently Mann
proved that, in a finite $p$-group, Hall's lower bound can be taken for at most
two distinct $d$. We improve this result and show that if $p$ is odd, then it
can only be taken for two distinct $d$ in a group with order $p^6$.
