Finite Group Databasebeta

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Cayley diagram

C22×#1843 #6371113

order512
nilpotency class2
isoclinismisoclinic class of #392
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC22×#1843

Subgroups

subgroupG/H
centerC24×C4C23
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socle, prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2);(3,4);(6,8)(10,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,6,7,8)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(9,10,11,12)(15,21)(18,24)(22,27)(25,28)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#21683, C2×#243908
maximal normal subgroup ofC22×#21683, C2×#243908

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278309);