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:
Why $G_n$'s are mutually non-isomorphic? (i.e., $G_n$ and $G_m$ are not isomorphic for all distinct natural numbers).
Could you please help me to find my answer.
Thanks,