On finite $p$ -group of class two with cyclic center

Question

Does there exist a finite $p$-group of nilpotentcy class $2$ such that

1. $Z(G)$ is not a subgroup of $\Phi(G)$, where $Z(G)$ is center of $G$ and $\Phi(G)$ is the Frattini subgroup of $G$.

2. $G/Z(G)$ is generated by 2 elements.

3. $Z(G)$ is cyclic.

Explanation:

For condition 1 and 2 we can say $D_{8}\times C_{2}.$

For condition 2 and 3 we can say $D_{8}$.

For condition 1 and 3 ?

For condition 1 and 2 and 3?

Thank you

Answer 1

The central product of $D_8$ and $C_4$ is such an example. More generally, for arbitrary primes $p$, you can take the central product of an extraspecial group of order $p^3$ with $C_{p^2}$.