Given a finite group $G$, the Class Equation can be gotten by considering the action of $G$ on itself by conjugation. If $H\le G$, then we can consider the action of $H$ on $G$ by conjugation, which -unless I'm mistaken- leads to the following orbit equation:
$$|G|=|C_G(H)|+\sum_{a\in \{Orbits'\space reps\}}\frac{|H|}{|C_G(a)\cap H|} \tag 1$$
where:
- "$Orbits$" (capital "O") stands for the orbits (under this action) of size bigger than $1$.
- $C_G(H)$ is the centralizer of $H$ in $G$.
If, in addition, $H\unlhd G$, then we can consider also the action of $G$ on $H$ by conjugation, which -again if I did things properly- leads to this other orbit equation:
$$|H|=|H \cap Z(G)|+\sum_{h \in \{Orbits \space rep's\}}\frac{|G|}{|C_G(h)|} \tag 2$$
where, again, "$Orbits$" (capital "O") stands for the orbits (under this action) of size bigger than $1$.
In spite of the popularity of the Class Equation, I never happened to see any utilization of its "variations" $(1)$ and $(2)$ (or the correct versions of them, if some mistake is there).
Is there any practical application of the orbit equations $(1)$ and $(2)$?
As an example of a "practical application" of the eq. $(2)$: if $G$ is a finite $p$-group and $H\unlhd G$ has order $p$, then $H\le Z(G)$.