Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups".
(i) Let $T$ be a finite $p$-group whose Frattini subgroup is cyclic and central. Then $T'$ has order $1$ or $p$ and there are subgroups $X, Y$ such that $T$ = $X\circ Y$ where $X$ is extraspecial and $Y$ has an abelian maximal subgroup and $\Omega_1(Y)$ is elementary abelian.
(ii) If $T/T'$ is elementary abelian, $Y$ is of the form $p^r$ or $\mathbb{Z}_{p^2} \times p^r$.
I've spent some time now on thinking of one example of $T$ such that its $Y $ is of the form $\mathbb{Z}_{p^2} \times p^r$. For $T$ the $Y$ of which of the former form, we have the extraspecial groups as examples, I suppose.