E₈(2) - Knowlpedia

For $q=2$ , $E_8(2)$ is the finite Chevalley central quotient of type $E_8$ :

E_8(2)=\mathbf G_{E_8,\mathrm{sc}}(\mathbb F_{2})/Z\bigl(\mathbf G_{E_8,\mathrm{sc}}(\mathbb F_{2})\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(2)|=337804753143634806261388190614085595079991692242467651576160959909068800000.

It is a nonabelian simple group. The exact order is 337,804,753,143,634,806,261,388,190,614,085,595,079,991,692,242,467,651,576,160,959,909,068,800,000.

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.
ATLAS of Finite Group Representations: E₈(2), Group heading: exact order 337804753143634806261388190614085595079991692242467651576160959909068800000.