TITLE: Bounds on the lowest dimension of a faithful representation of a finite group
QUESTION [12 upvotes]: Let $G$ be a finite group. Can one always construct a faithful representation of $G$ over $\mathbb C$ of dimension $\le \sqrt{|G|}$? What would be the order of the minimal such dimension for a "random" group? Are the worst groups (with the largest minimal faithful dimensions) special in some way?

REPLY [10 votes]: I'm writing a paper where I prove that all counterexamples are related to Derek Holt's example: if G is a finite group then either G has a faithful representation over $\mathbb{C}$ of dimension $\leq\sqrt{|G|}$ or $G$ is a $2$-group with center is elementary abelian of order 8 and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ vanish on $G-Z(G)$.
For any of these $2$-groups, the minimal dimension of a faithful representation is $\frac{3}{\sqrt{8}}\sqrt{|G|}$.
I also prove that this minimal dimension is equal to $\sqrt{|G|}$ if and only if $G$ is a $2$-group with center elementary abelian of order either $4$ or $16$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ vanish on $G-Z(G)$.
I'd be happy to share my preprint. It would help me to know the motivation for this question.