Let $G$ be a finite group so that $\frac{G}{Z(G)}\cong \mathbb{Z}_p \times \mathbb{Z}_p$, where $p$ is a prime. Then $\frac{G}{Z(G)}$ has the presentation $\langle aZ(G),bZ(G)|a^p ,b^p , aba^{-1}b^{-1}\in Z(G)\rangle$.
I want to determine the centralizer $C_G (g)$ of an arbitrary element $g\in G\setminus Z(G)$ in term of cosets of $Z(G)$ in $C_G (g)$, exactly.
For doing this, I take $g\in G\setminus Z(G)$. Then $Z(G)\neq gZ(G)\in \frac{G}{Z(G)}$. Every element of $\frac{G}{Z(G)}$ has the form $a^i b^j Z(G)$ for some $1\leq i\leq p$ and $1\leq j\leq p$. Especially, I assume that $gZ(G)=a^i Z(G)$ for some $1\leq i\leq p$. So $g=a^i z$ for some $z\in Z(G)$. Then $C_G (g)=C_G (a^i z)=C_G (a^i)$. Now it is enough to compute $C_{G}(a^i )$ in term of cosets of $Z(G)$ in $C_G (a^i )$. I know that $C_G (a^i )=Z(G)\sqcup \Big(\bigsqcup_{t\in C_G (a^i)\setminus Z(G)}tZ(G)\Big)$. But I don't know for which $t$ the equation holds.
Thanks in advance.