The answer is provided by the following
Prill's isomorphism criterion. Let $G_1, G_2 \subset GL(n, \textrm{GL}(n, \mathbb{C})$ be two finite subgroups, $n \geq 2$. Assume moreover that they are small, i.e. without pseudo-reflections. Then the two germs of quotient singularities $$(\mathbb{C}^n/G_1, 0), \quad (\mathbb{C}^n/G_2, 0)$$ are analitically isomorphic if and only if $G_1$ and $G_2$ are conjugated in $GL(n, \textrm{GL}(n, \mathbb{C})$.
The reference is
D. Prill: Local classification of quotients of complex manifolds by discontinuous groups, Duke Mathematical Journal 34 Number 2 (1967), 375-386.
See also this paper, Theorem 1.2 page 126.
