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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.

The answer is provided by the following

Prill's isomorphism criterion. Let $G_1, G_2 \subset GL(n, \mathbb{C})$ 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 $$(\mathbb{C}/G_1, (\mathbb{C}^n/G_1, 0), \quad (\mathbb{C}/G_2, \mathbb{C}^n/G_2, 0)$$ are analitically isomorphic if and only if $G_1$ and $G_2$ are conjugated in $GL(n, \mathbb{C})$.

The reference is

D. Prill: Local classification of quotients of complex manifolds by discontinuous groups, Duke Mathematical Journal 34 Number 2 (1967), 375-386.

Prill's isomorphism criterion. Let $G_1, G_2 \subset GL(n, \mathbb{C})$ be two finite subgroups that are small, i.e. without pseudo-reflections. Then the two germs of quotient singularities $$(\mathbb{C}/G_1, 0), \quad (\mathbb{C}/G_2, 0)$$ are analitically isomorphic if and only if $G_1$ and $G_2$ are conjugated in $GL(n, \mathbb{C})$.