Number of prime divisors of element orders from character table. From wikipedia:

It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of Graham Higman).

Precisely how is this done?  I would be happy with a reference.  (In particular, I am looking for the location of the elements in a solvable group with which have the maximum number of prime divisors in their order.  If somebody has any extra information on that particular situation that would also be appreciated.)

A formula for the order of each conjugacy class of a finite group is given in Exercise 2.21 of Fulton and Harris Representation Theory: A first Course; namely $c(g) = {|G| \over \sum_{\chi} \overline{\chi(g)}\chi(g)}$ where the sum is over the irreducible representations, and $c(g)$ is the number of elements conjugate to $g$.
This isn't exactly what you asked for, but in any particular case you could calculate the prime divisors of the orders of the elements of each conjugacy class from this by factorisation.
