Finite Group Databasebeta

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Cayley diagram

C4×#2685 #521184

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC4×#2685

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

4180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72304109
maximal normal subgroup of#72304109

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428380);