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Let $G$ be finite p-group of nilpotency class n and $d(G)=2$ with cyclic center $Z(G)=\langle z\rangle$ and $Z_2(G)$ is non-cyclic abelian group of order $p^3$
I want to understand relation between upper central series and lower central series
I know because class of $G$ is n then $Z_n(G)=G$ and $\gamma_{n+1} (G)=1$ since Z(G) is cyclic then for some $i$ we must have $Z(G) \le \gamma_i(G)$
Is there any relation to find this $i$?
I wonder how coclass of group effect upper central series and lower central series? for example if we know coclass $G$ is of coclss 2, what this tell us?

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  • $\begingroup$ Clearly $Z(G) \le \gamma_1(G)=G$, but it is not true in general that $Z(G) \le \gamma_2(G)$. $\endgroup$ – Derek Holt Sep 6 '15 at 20:08
  • $\begingroup$ what property $G$ must have to $Z(G)\le \gamma_2 (G)$ $\endgroup$ – user148528 Sep 7 '15 at 6:11
  • $\begingroup$ It is impossible to answer that without having some idea of what kind of property you are thinking of. $\endgroup$ – Derek Holt Sep 7 '15 at 13:54
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    $\begingroup$ I added some property, I hope they are helpful. $\endgroup$ – user148528 Sep 7 '15 at 18:57

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