Finite Group Databasebeta

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Cayley diagram

C2×#20830 #6371198

order512
nilpotency class3
isoclinismisoclinic class of #3055
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#20830

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C4⋊C4
prime radicalC25C2×Q8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2262 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)(11,12,15,18,16,19,14,24)(13,20,23,25,21,26,22,17);(4,7)(8,10);(11,13,14,22,16,21,15,23)(12,17,18,20,19,25,24,26);(3,5,6,9)(4,8,7,10)(13,21)(17,25)(20,26)(22,23)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238763, #72238765, #72238768, …
maximal normal subgroup of#72238763, #72238765, #72238768, …
maximal quotient of#72239131, #72323061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278394);