  Let $p$ be a an odd prime and let $G$ be a finite $p$-group with cyclic
commutator subgroup $G'$. We prove that the exponent and the abelianization of
the centralizer of $G'$ in $G$ are determined by the group algebra of $G$ over
any field of characteristic $p$. If, additionally, $G$ is $2$-generated then
almost all the numerical invariants determining $G$ up to isomorphism are
determined by the same group algebras; as a consequence the isomorphism type of
the centralizer of $G'$ is determined. These claims are known to be false for
$p=2$.
