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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 $Z_{n-3}(G)$?

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  • $\begingroup$ The conditions 4, 6, 7, can be deduced from 1 and 5. Also if $G$ is not regular then $cl(G) \geq p$. I suppose that $n$ is defined by $p^n=|G|$, if so then $G$ has class $n-2$; however if $\Phi(G)=Z_{n-4}$, then $Z_{n-3}(G)=G$ which means that $G$ has class $n-3$. I wish that you edit your question with this remarks in mind. $\endgroup$
    – Yassine Guerboussa
    Commented May 6, 2014 at 14:18
  • $\begingroup$ Now I suppose that you mean $\Phi(G)=Z_{n-3}(G)$ in your second question. I think yes, $G/Z_{n-3}$ has order $p^2$ and it is not cyclic, thus it contains $\Phi(G)$. Since $G$ is 2-generated, it follows that $\Phi(G)=Z_{n-3}(G)$. $\endgroup$
    – Yassine Guerboussa
    Commented May 6, 2014 at 14:27
  • $\begingroup$ @YassineGuerboussa thank you for your answers. About the Frattini subgroup, the answer is very easy, i should not ask for. Instead i've no answers about $\Omega_{1}(Z_{2}(G))$. $\endgroup$
    – Marco Ruscitti
    Commented May 7, 2014 at 10:26

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