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Cayley diagram

C4×#2635 #515195

order512
nilpotency class2
isoclinismisoclinic class of #2613
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC4×#2635

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#2635C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

3810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72297765, #72297767, #72297770
maximal normal subgroup of#72297765, #72297767, #72297770
maximal quotient ofC4×#14722

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422391);