Let $F$ be a finite group.

Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially?

For example the Bar construction has the property that there are exactly $(|F|-1)^n$ nondegenerate $n$-simplices. This answers the question affirmatively for $\mathbb{Z}/2$, but for other groups it still grows exponentially.

A lower bound for the number of such simplices is of course given by the rank of the group homology and in all examples that I know this only grows polynomially.

Of course it would be nice to have a functorial model, but that might be a follow up.