TITLE: Counting conjugacy classes in simple groups of Lie type
QUESTION [11 upvotes]: Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}_q$ of characteristic $p$: these are split or quasi-split.   There are also several families obtained less directly from algebraic groups: the Suzuki groups $^2\!B_2(q)$ and the Ree groups $^2\!G_2(q),\: ^2\!F_4(q)$, where $q = p^{2n+1}$ with $n \geq 1$ is an odd power of (respectively) $2,3,2$.  [Caution: Sometimes $q$ is written here as $q^2$ to emphasize the similarity of group orders to those of untwisted groups.]
Counting the total number of conjugacy classes in such a group is a natural problem, relative to the determination of ordinary characters.  Case-by-case study has been done for many of the families, especially the exceptional types; but it's unclear how much can be expressed uniformly.   (Older results are surveyed in Chapter 8 of my 1995 AMS book on conjugacy classes.)    It helps to assume the ambient algebraic group is simply connected, in which case Steinberg proved for split and quasi-split types that the number of semisimple classes (whose elements have order prime to $p$) is $p^r$ with $r$ the Lie rank.   For Suzuki or Ree groups, $r$ is the rank of the BN-pair: 1, 1, 2.
For Suzuki groups the total number of classes is $q + 3$ (Deriziotis), while for Ree groups of type $G_2$ the number is $q+8$ (Ward).  The latter groups came up recently here.

How many conjugacy classes do the Ree groups of type $F_4$ have?

This should be computable from known data, though not easily.  Judging from all special cases I'm aware of, the answer should be given by a polynomial $q^2 + aq + b$, where $a, b \in \mathbb{Z}$ are independent of $q$.  [Note however that for most other families of groups of Lie type, there are extra complications related to isogeny type, bad primes, and such.]
The underlying question, of course, is whether one can predict a priori what the polynomials will look like for all groups (coming from simply connected algebraic groups) starting with the highest degree term $q^r$.

REPLY [8 votes]: According to

Fleischmann, Peter; Janiszczak, Ingo.
  On the computation of conjugacy classes of Chevalley groups.
  Appl. Alg. in Eng., Comm. and Comp. 1996, 7(3), 221--234

the class number of ${\rm F}_4(q)$ is

$q^4 + 2q^3 + 6q^2 + 10q + 19$ if $q = 2^n$,
$q^4 + 2q^3 + 7q^2 + 15q + 30$ if $q = 3^n$, and
$q^4 + 2q^3 + 7q^2 + 15q + 31$ if $q = p^n$ where $p > 3$

(see page 233).
According to Frank Lübeck's database on finite groups of Lie type,
the class number of $^2{\rm F}_4(q^2)$ is $q^4+4q^2+17$.
In that database you also find class numbers for many other types of finite
groups of Lie type.