The group $G$ is generated by the two elements $\sigma$ and $\tau$, of order $5$ and $4$ respectively. We assume that $\tau\sigma\tau^{-1}=\sigma^2$.

I have shown the following:  

\* $\tau\sigma^k\tau^{-1}=\sigma^{2k}$ and $\tau^k\sigma\tau^{-k}=\sigma^{2^k}$.  

\* $\langle\sigma\rangle$ is a normal subgroup of $G$, and $\langle\sigma\rangle\cap\langle\tau\rangle=\{e\}$.  

\* $G/\langle\sigma\rangle=\langle\tau\langle\sigma\rangle\rangle$.  

\* $G$ is of order $20$ and every element $g$ in $G$ may be written uniquely in the form $g=\sigma^k\tau^m$, where $0\le k<5$ and $0\le m<4$.  

\* The commutator subgroup $[G:G]=\langle\sigma\rangle$.

What remains is to find the center $Z(G)$ of $G$. Any suggestions on how to proceed? Thank you.