What is a non-alternating simple group with big order, but relatively few conjugacy classes?

Question

I'm not sure if this question is legal.

I'm writing my BsC thesis on character theoretical calculations and I have already calculated a lot of character tables (a few alternating, symmetric, $\operatorname{SL}(2,3)$, etc..), but I still have some space, so I thought I could choose a group, with not too many conjugacy classes (around $7$ still fit in nicely), that is preferably not monomial.

What do you suggest?

Thanks in advance!

Answer 1

I'm a big fan of the group $\operatorname{PSL}(2,7)$, also known as $\operatorname{GL}(3,2)$, which is finite, simple, not isomorphic to an alternating group, and has precisely $6$ conjugacy classes.