PSL(2,8) - Knowlpedia

$\operatorname{PSL}_{2}(8)$ is the following finite group, with $n=2$ and $q=8$ :

\operatorname{PSL}_{2}(8)=\operatorname{SL}_{2}(\mathbb F_{8})/\{\lambda I:\lambda\in\mathbb F_{8}^\times,\ \lambda^{2}=1\}, \qquad \operatorname{SL}_{2}(\mathbb F_{8})=\{A\in M_{2}(\mathbb F_{8}):\det A=1\}.

This is the projective special linear group; multiplication is multiplication of scalar cosets. Here $\mathbb F_{8}$ is the field with $8$ elements.

Order and simplicity

Its order is

|\operatorname{PSL}_{2}(8)|=504.

It is a nonabelian simple group. The exact order is 504.

Parameter convention

Type $A_{n-1}$ uses matrix size $n$ , so the Lie rank is $n-1$ . The simple family omits $\operatorname{PSL}_2(2)\cong S_3$ and $\operatorname{PSL}_2(3)\cong A_4$ .

References

Peter J. Cameron, Notes on Classical Groups, §§2.4–2.5, pp. 19–21; Theorem 2.11.