Projective special linear group:PSL(2,8)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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