What finite subgroup $G$ of $SU(N)$ most closely approximates $SU(N)$?
I'm hoping that the answer isn't too dependent on the precise definition of "approximates," but I'll offer a definition anyway to make the question well-posed: consider the fundamental representation of $SU(N)$ by $N\times N$ unitary matrices, and define the distance between two matrices $A,B$ by $$ |A-B|=\frac{1}{N}\text{trace}\big((A-B)^\dagger (A-B)\big). \tag{1} $$ The error in the approximation can be quantified by $$ \max_{u\in SU(N)} \min_{g\in G} |u-g|. \tag{2} $$ The normalization of (1) is meant to give a more fair comparison between different values of $N$.
There is a positive lower bound on the approximation of $SU(N)$ by its finite subgroup (for $N\ge 2$) due to Jordan's Lemma on finite subgroups of $SU(N)$: Every finite subgroup contains an abelian subgroup of index $\le q(N)$. See
Turing, A. M., Finite approximations to Lie groups, Ann. Math. (2) 39, 105-111 (1938). ZBL0018.29801.
As for an optimal finite subgroup, good luck with that, this sounds like one of the sphere packing problems which are notoriously difficult. Even optimal values of $q(N)$ in Jordan's theorem are still unknown although much work was done in this direction.