Finite Group Databasebeta

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Cayley diagram

C2×#28479 #6465625

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#28479

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6465638, C2×#6574138, C2×#6578332
maximal normal subgroup ofC2×#6465638, C2×#6574138, C2×#6578332

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372821);