For a finite group $G$ denote by $\operatorname{conj}\left(G\right)$ the number of conjugacy classes of $G$ and let $q\left(G\right)$ be the quotient:
\begin{align} q\left(G\right)=\frac{\operatorname{conj}\left(G\right)}{|G|} \end{align}
It's obvious that $0<q\left(G\right) \le 1$ and it's $1$ iff $G$ is abelian. Obviously the number $q\left(G\right)$ is "small" when $\operatorname{conj}\left(G\right)$ is small with respect to $|G|$.
Informally: The number $q\left(G\right)$ is small when the conjugacy classes are big for a lot of $g\in G$. This is equivalent to say that the stabilizer $C_G\left(g\right)$ is small (has a big index) for a lot of $g\in G$. In particular the center $Z\left(G\right)$ has small index.
I'm looking for families of examples of non abelian finite Coxeter groups having small $q\left(G\right)$. Intuitively I think that for Coxeter groups this numbers in general is "small". (for me: the smaller the better).
One example: Let $G= S_n$ the symmetric group then $q \left(S_n \right)= p\left(n \right) / n!$ where $p\left(n \right)$ denotes the partitions of $n$. Note that $q\left(S_n \right)\to 0$.
By small I don't require some condition like the above $q\left(S_n \right)\to 0$ it's up to you what it means by small and I'll see if it's useful. For example If you know that $G_i$ is a family of Coxeter groups with $q\left(Q_i \right)< 1/3$ that's good and if other family has an upper bound of $1/10$ that's even good.
If someone knows some results about the number of conjugacy classes for Coxeter groups or at least some boundaries for the index of the stabilizers I would really appreciate it.