$G$ is a finite group generated by two elements $a$ and $b$, we are given the following data:
Order of a= $2$
Order of $b=2$
Order of $ab=8$.
If $Z(G)$ denotes the center then what is $G/Z(G)$ isomorphic to?
To be honest I don't know how to start this. I thought of taking $D_8$ as a concrete example of such a group but was not able to proceed much.
One thing I can see that this group is non abelian as if it was abelian then it would imply that $(ab)^2=e$ which contradicts the fact that order of $ab$ is 8.
What can I say more about this group?