I have two questions about a specific type of finite $p$-groups that i've seen in an interesting work on automorphisms of finite p-groups.

Let $G$ a finite $p$-group of order $p^{n}$ such that:

- $G$ is two generated $p$-group of coclass $2$ and $p \neq 2$;

- $G$ is not regular, $G$ is not powerful;

- $C_{G}(Z(\Phi(G)) = \Phi(G)$;

- $Z_{2}(G) / Z(G) \cong C_{p} \times C_{p}$.

From these assumptions we have that $cl(G) \geq p$; $Z(G) \cong C_{p}$; $Z_{3}(G)/Z_{2}(G)\cong C_{p}$; $G/Z_{2}(G)$ is of maximal class.

So if one looks to the upper central series this group is almost a maximal class group except for the quotient $Z_{2}(G)/Z(G)$.

Is it true that $\Omega_{1}(Z_{2}(G))$ is contained in $\gamma_{3}(G)$?

Is it true that the Frattini subgroup coincides with $Z_{n-3}(G)$?