Finite Group Databasebeta

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Cayley diagram

C42×#126 #513376

order512
nilpotency class2
isoclinismisoclinic class of #120
automorphism?
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4

Aliases

direct productC42×#126

Subgroups

subgroupG/H
centerC22×C42C23
commutatorC22C23×C42
Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C42 ◁ G
LECSG ▷ C24 ▷ C1

3750 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)(11,16)(12,14)(13,15)(17,18)(19,21)(20,22)(23,24);(9,11,13,12)(10,14,15,16)(17,19,20,23)(18,21,22,24);(9,12,13,11)(10,14,15,16)(18,22)(21,24)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72228083, C42×#126⋊C3, #126×C42⋊C3, …
maximal subgroup of#72228083, C42×#126⋊C3, #126×C42⋊C3, …
maximal normal subgroup of#72228083, C42×#126⋊C3, #126×C42⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420572);