throughout a research problem about finite $p$-groups, I have a challenge as follows,
Let $G$ be a finite non-abelian $p$-group, where $p$ is odd and $Z(G)$ is non-cyclic. ($Z(G)$ denotes the center of $G$). Also let $G$ satisfies the following two properties
(a) There exists a minimal normal subgroup $N$ of $G$ such that for any $x\in G$, if $\langle x\rangle \ntriangleleft G$, then $\langle x,N\rangle \ntriangleleft G$.
(b) For any minimal normal subgroup $V$ of $G$ and the subgroup $U$ of $G$, where $\displaystyle\frac{U}{V}:=Z(\frac{G}{V})$, any cyclic subgroup of $U$ is normal in $G$ and $\exp(U)=p^2$.
Question Does there exists a minimal normal subgroup $T$ of $G$, such that $\displaystyle Z(\frac{G}{T})=\frac{Z(G)}{T}$?
Some examples in GAP library, show that may be the answer is yes, even if for $p$-groups, which only satisfies the property (a).
(With a wide approach and independent of any property, I wonder if there exists any information or
classification of finite non-abelian $p$-groups with a minimal normal subgroup,
which satisfies the property in my question.)
Update: After an attempt I find another restriction on the subgroups $U$ as $\exp(U)=p^2$. Now according to the examples in GAP library, may be the minimal normal subgroup whose generator is the product of all generators of minimal normal subgroups of $G$ could have the desired role in my question.
Many thanks for your time on my question. Any Answer or comment will be greatly appreciated.
