E₆(2) - Knowlpedia

For $q=2$ , $E_6(2)$ is the finite Chevalley central quotient of type $E_6$ :

E_6(2)=\mathbf G_{E_6,\mathrm{sc}}(\mathbb F_{2})/Z\bigl(\mathbf G_{E_6,\mathrm{sc}}(\mathbb F_{2})\bigr).

Here $\mathbf G_{E_6,\mathrm{sc}}$ 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_6(2)|=214841575522005575270400.

It is a nonabelian simple group. The exact order is 214,841,575,522,005,575,270,400.

Rank and central quotient

The absolute rank is $6$ . The finite center of the simply connected group has order $\gcd(3,q-1)$ ; 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.