Let \(G\) be a simple, algebraic group of exceptional type \(E_ 8\) over an algebraically closed field \(K\) of characteristic \(p > 0\) and defined over the finite field \(\mathbb{F}_ q\), where \(q\) is a power of \(p\). Let \(F\) be the corresponding Frobenius morphism of \(G\), and \(G^ F\) the fixed point group, which is the finite simple group of Lie type \(E_ 8\). The authors determine the semisimple conjugacy classes of \(G^ F\) as well as the total number of conjugacy classes, given as a polynomial in \(q\).
The corresponding results for adjoint groups of type \(E_ 6\) and \(E_ 7\) can be found in the authors’ paper [Commun. Algebra 21, 93-161 (1993; Zbl 0813.20015)].