Finite Group Databasebeta

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Cayley diagram

C4×#2110 #492831

order512
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphism?
homologyAbelianization (H1) = C2×C43 · Schur multiplier (H2) = C23×C42

Aliases

direct productC4×#2110

Subgroups

subgroupG/H
centerC4×C8C42
commutatorC4C2×C43
Frattini, cosocleC22×C8C24
socleC22C22×#97
prime radicalC24C2×M16
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4);(5,6,8,12,9,14,20,28,10,13,21,27,22,29,35,36)(7,15,17,31,18,24,33,26,16,30,19,34,32,11,23,25);(5,7)(6,11,13,24)(8,19)(9,18)(10,16)(12,26,27,25)(14,15,29,30)(17,21)(20,23)(22,32)(28,34,36,31)(33,35);(7,16)(11,24)(15,30)(17,19)(18,32)(23,33)(25,26)(31,34)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227009, C4×#33329, #72227060, …
maximal normal subgroup of#72227009, C4×#33329, #72227060, …
maximal quotient of#72227009, #72227016

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400027);