Finite Group Databasebeta

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Cayley diagram

C4×#2578 #514968

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27

Aliases

direct productC4×#2578

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

2966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72304042, #72304102
maximal normal subgroup of#72304042, #72304102

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422164);