Finite Group Databasebeta

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Cayley diagram

C22⋊C4×#131 #6369872

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4

Aliases

direct productC22⋊C4×#131

Subgroups

subgroupG/H
centerC22×C8C24
commutatorC22C23×C42
Frattini, cosocleC22×C4C25
socleC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5274 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:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22⋊C4×#576, #72284641, #72285212
maximal normal subgroup ofC22⋊C4×#576, #72284641, #72285212

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277068);