Finite Group Databasebeta

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Cayley diagram

C2×#21370 #6372327

order512
nilpotency class3
isoclinismisoclinic class of #447
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#21370

Subgroups

subgroupG/H
centerC2×C42C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C4⋊C4
prime radicalC25C2×Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240023, C2×#6375061
maximal normal subgroup of#72240023, C2×#6375061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279523);