E₈(q) - Knowlpedia

For a prime power $q\geq 2$ , $E_8(q)$ is the finite Chevalley central quotient of type $E_8$ :

E_8(q)=\mathbf G_{E_8,\mathrm{sc}}(\mathbb F_{q})/Z\bigl(\mathbf G_{E_8,\mathrm{sc}}(\mathbb F_{q})\bigr).

Here $\mathbf G_{E_8,\mathrm{sc}}$ is the split simply connected algebraic group of the displayed root-system type. Its finite points are fixed by coordinatewise $q$ -power Frobenius, and multiplication is multiplication of cosets of their finite center.

Order and simplicity

Its order is

|E_8(q)|=q^{120}(q^{2}-1)(q^{8}-1)(q^{12}-1)(q^{14}-1)(q^{18}-1)(q^{20}-1)(q^{24}-1)(q^{30}-1).

It is a nonabelian simple group for every admitted prime power q.

Rank and central quotient

The absolute rank is $8$ . The finite center of the simply connected group has order $1$ ; the notation here always denotes the quotient by that center. There are no omitted small-field simplicity exceptions for this type.

References

Robert Steinberg, Lectures on Chevalley Groups (Yale, 1967), §3 construction; §4 Theorem 5, PDF pp. 52–54; §9 Theorems 24–25, PDF pp. 136–138.
B. Akbari, ODs-characterization of some low-dimensional finite classical groups, Table 2, p. 85: order column only; its additional Restrictions concern a separate invariant.