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](https://mathoverflow.net/questions/132116/).



> 
> 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 $q^r$. 



Counting conjugacy classes in simple groups of Lie type
According to



> 
> Fleischmann, Peter; Janiszczak, Ingo.
>  [On the computation of conjugacy classes of Chevalley groups](http://link.springer.com/article/10.1007%2FBF01190331).
>  *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](http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/index.html?LANG=de) 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.


