Finite Group Databasebeta

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Cayley diagram

C22×#2371 #7625369

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC22×#2371

Subgroups

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

8674 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);(5,6,8,7)(10,13)(14,16)(17,18)(19,22)(20,21)(23,24);(5,7,8,6)(18,21)(22,24);(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,22,21,24);(9,11,12,15)(10,14,13,16)(17,18)(19,22)(20,21)(23,24)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#44250, C22×#34450
maximal normal subgroup ofC22×#44250, C22×#34450

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532565);