This question was answered by @Jim Belk And he defined $G_n$ as follows: $$ G_n \;=\; \langle a,b \mid [a^{-1}ba,b] = \cdots = [a^{-n}ba^n,b]=1\rangle $$ My question is:
Are $G_n$'s mutually non-isomorphic? (i.e., for every two distinct natural numbers $n$ and $m$, $G_n$ and $G_m$ are not isomorphic).

Could you please help me to find my answer.

Thanks,