Finite Group Databasebeta

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Cayley diagram

C4×#2487 #514934

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC4×#2487

Subgroups

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

3850 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);(5,6)(7,8)(9,10,13,16)(11,15,12,14)(17,18,20,22)(19,21,23,24);(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,23)(22,24);(6,8)(19,23)(21,24);(9,12,13,11)(10,15,16,14)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264166, #72264221, #72315931, …
maximal normal subgroup of#72264166, #72264221, #72315931, …
maximal quotient of#72265431

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422130);