on relation between upper central series and lower central series

Question

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?