Finite Group Databasebeta

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Cayley diagram

C2×#11341 #358488

order512
nilpotency class3
isoclinismisoclinic class of #2002
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11341

Subgroups

subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23#385
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214172, #72215689, C2×#379089, …
maximal normal subgroup of#72214172, #72215689, C2×#379089, …
maximal quotient of#72214091, #72214132, C2×#123973

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265684);