The problem is inspired by eigenvalue bounds of random Cayley graphs on $SL_2(q)$.

**Definition.** An infinite series of finite groups $S$ is **α-rich** if the dimension of the smallest nontrivial representation of $G$ on $\mathbb{C}$ is $\Omega(|G|^\alpha)$ for every $G\in S$.

For example, the series of groups $SL_2(q)$ is $\frac{1}{3}$-rich.

**Question.** Are there any series of $α$-rich groups with $\alpha>\frac13$?

**Known.** One only needs to consider simple groups, and by CFSG, it's just going through Lie groups.

Along the lines of David E Speyer, a permutation representation on $O(|G|^\frac13)$ points rules out the possibility for the group to beat $SL_2(q)$.

This holds for all classic groups of type $A_n (n\geq2)$, $B_n(n\geq3)$, $C_n(n\geq2)$, $D_n(n\geq4)$, $^2D_n(n\geq3)$, $^2A_n(n\geq4)$: just consider the action of these groups on the 1-dimensional subspaces of their defining vector space. $^2A_2$ is not in the list, but $^2A_2(q)$ has a representation on $q^2-q+1$ dimensions, hence ruled out.

The same argument rules out $F_4$, $E_n$ and $^2E_6$: All of them are covered in a paper given by Derek Holt. None of them beats $SL_2(q)$.

$G_2$ and $^2G_2$ do not beat $SL_2(q)$, by Derek Holt's answer.

For $^2B_2(q)$, see Wikipedia: it almost beats $SL_2$, but it has two characters of dimension $O(q^{3/2})$.

For $^3D_4(q)$, see Deriziotis, D. I., and G. O. Michler. "Character table and blocks of finite simple triality groups $^3D_4(q)$." Transactions of the American Mathematical Society 303.1 (1987): 39-70.: there's a character of dimension $q(q^4-q^2+1)$.

The last case $^2F_4$ is found in Die unipotenten Charaktere fur die GAP-Charaktertafeln der endlichen Gruppen vom Lie-Typ. M. Claßen-Houben; Diplomarbeit, RWTH Aachen; 2005. According to the conventions of the paper, the group has size $q^{52}$, and there's a character of dimension $q^2Φ_{12}Φ_{24}$, which is $O(q^{14})$.