  Let $G$ be a finite group, $p$ a prime, and $IBr_p(G)$ the set of irreducible
$p$-Brauer characters of $G$. Let $\bar e_p(G)$ be the largest integer such
that $p^{\bar e_p(G)}$ divides $\chi(1)$ for some $\chi \in IBr_p(G)$. We show
that $|G:O_p(G)|_p \leq p^{k \bar e_p(G)}$ for an explicitly given constant
$k$. We also study the analogous problem for the $p$-parts of the conjugacy
class sizes of $p$-regular elements of finite groups.
