Finite Group Databasebeta

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Cayley diagram

C22×#2382 #7625925

order512
nilpotency class2
isoclinismisoclinic class of #560
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC22×#2382

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutatorC22C25×C4
Frattini, cosocleC23C26
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C23 ▷ C1

7890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2);(3,4);(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(18,19)(22,23);(10,13)(11,16)(17,18,21,19)(20,23,24,22);(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(9,11,12,16)(10,14,13,15)(17,20,21,24)(18,22,19,23)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#34450, C22×#50644
maximal normal subgroup ofC22×#34450, C22×#50644

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533121);