# Existence of a special Central cyclic subgroup

I have two related questions. Let $G$ be a finite $p$ group. Can we always choose a central subgroup $N$ of order $p$ not contained in the commutator subgroup? Clearly we cannot do that for extra special $p$ groups.

We can always choose a central subgroup $N$ of order $p$ in a finite nilpotent group as well. Is something similar possible for finite solvable groups?

For example for the first, in the generalized quaternion groups, there is a unique element of order $2$, which is thus contained in any non-trivial subgroup.
For the second, consider either of the groups the group $S_3$ or $S_4$ which are solvable but have trivial centers.
• But i hope the class of $p$ groups($p$ odd) for which we can choose a cyclic central subgroup which is not contained in the commutator is large.