This question didn't receive an answer on MathSE, so I'm asking it here.

Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic zero. Every $1$-dimensional representation of $G$ over $k$ factors through $G^{\mathrm{ab}} = G/[G, G]$, and every finite abelian group has a faithful representation over $k$. Taken together, these simple facts show that one can characterise the abelianisation $G^{\mathrm{ab}} = G/[G, G]$ of $G$ as the smallest quotient of $G$ such that every $1$-dimensional representation of $G$ factors through it.

Let $G_n$ be the smallest quotient of $G$ such that every $n$-dimensional representation of $G$ over $k$ factors through $G_n$. Clearly $G_1 = G^{\mathrm{ab}}$ and I've read that if $G$ has a generating set of size $m$ then $G_m = G$, i.e. $G$ has an $m$-dimensional faithful representation. What do the $G_n$ look like in general? Can they be characterised in a different way? What do they tell us about $G$?

This question didn't receive an answer on MathSE, so I'm asking it here.

Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic zero. Every $1$-dimensional representation of $G$ over $k$ factors through $G^{\mathrm{ab}} = G/[G, G]$, and every finite abelian group has a faithful representation over $k$. Taken together, these simple facts show that one can characterise the abelianisation $G^{\mathrm{ab}} = G/[G, G]$ of $G$ as the smallest quotient of $G$ such that every $1$-dimensional representation of $G$ factors through it.

Let $G_n$ be the smallest quotient of $G$ such that every $n$-dimensional representation of $G$ over $k$ factors through $G_n$. Clearly $G_1 = G^{\mathrm{ab}}$ and I've read that if $G$ has a generating set of size $m$ then $G_m = G$, i.e. $G$ has an $m$-dimensional faithful representation. What do the $G_n$ look like in general? Can they be characterised in a different way? What do they tell us about $G$?