# Projective special linear group:PSL(2,8)

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## Definition

This group is defined is defined in the following equivalent ways:

1. It is the projective special linear group of degree two over field:F8, the field of eight elements. It is denoted $PSL(2,8)$.
2. It is the projective general linear group of degree two over field:F8, the field of eight elements. It is denoted $PGL(2,8)$.
3. It is the special linear group of degree two over field:F8, the field of eight elements. It is denoted $SL(2,8)$.

### Equivalence of definitions

The equivalence of definitions follows from isomorphism between linear groups when degree power map is bijective.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 504 groups with same order As $\! SL(2,q), q = 8$: $\! q^3 - q = q(q-1)(q + 1) = 8(7)(9) = 504$
exponent of a group 126 groups with same order and exponent of a group | groups with same exponent of a group

### Arithmetic functions of a counting nature

Function Value Explanation
number of conjugacy classes 9 As $SL(2,q)$, $q = 8$ is a power of $2$: $q + 1 = 8 + 1 = 9$. Note that the formula is different for odd $q$. See element structure of special linear group of degree two
number of conjugacy classes of subgroups 12
number of subgroups 386

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group, simple non-abelian group Yes projective special linear group is simple (with a small list of exceptions)
minimal simple group Yes
Hurwitz group Yes It is the conformal automorphism group of the Macbeath surface, which has genus 7, and the order of the group is $84(7 - 1) = 84(6) = 504$, so it is a Hurwitz group.

## GAP implementation

### Group ID

This finite group has order 504 and has ID 156 among the groups of order 504 in GAP's SmallGroup library. For context, there are 202 groups of order 504. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(504,156)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(504,156);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [504,156]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description Functions used
PSL(2,8) PSL
PGL(2,8) PGL
SL(2,8) SL