I don't know much about (algebraic/etale) fundamental groups, so sorry if this question sounds stupid. I am interested in quotient singularities (quotients $X$ of $\mathbb{C}^n$ by a finite subgroup $G$ of $GL(n,\mathbb{C})$).

The divisor class group $Cl(X)$ of $X$ is the abelianization $G/[G,G]$ of $G$. Is there also a connection with the fundamental group (defined in any way)? Maybe even $G=\pi_1(X)$, as stated in a similar way here: https://mathoverflow.net/a/50638. I am not sure if my case fits in this setting. I would be very happy with references, since I did not find any convenient papers myself.