Helmut Wielandt [discussed][1] an old question (Chap. 2, Section 15, which can be dated back to Camille Jordan):

Let $g\neq 1$ be a permutation in some finite primitive permutation group $G$ of degree $n$. The minimal degree $m$ is defined to be the least number of points that $g$ permutes. It is known that if $m>3$, then $n$ is bounded by $$\frac{m^2}{4}\log\frac{m}{2}+m\left(\log\frac{m}{2}+\frac{3}{2}\right).$$

**Question:** Do we know a better upper bound today (with CFSG and O'Nan-Scott etc.)?


  [1]: https://books.google.com/books/about/Finite_Permutation_Groups.html?id=npviBQAAQBAJ