Abelianization of a $p$ group.

Question

Let $G$ be a $d$ generated finite $p$ group. Let $N$ be normal subgroup of order $p$ contained in $[G,G]\cap Z(G)$. Can we say say that $(G/N)/[G/N,G/N]=(G/N)_{ab}$ is a direct product of $d$ cyclic groups?

Answer 1

Clearly the factor group you give is abelian and thus the direct product of cyclic groups and Derek Holt's comment gives you that you do not need more than $d$.

If $d$ is also the minimal number of generators, Burnsides basis theorem shows that for $M=[G,G]G^p$ the group $G/M$ is the direct product of $d$ cyclic groups of order $p$. Since your $N\le M$, this shows that $d$ also is the minimal cardinality of a generating set for $G/N$.