Finite Group Databasebeta

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Cayley diagram

C22⋊C4×#126 #519475

order512
nilpotency class2
isoclinismisoclinic class of D4×#120
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC22⋊C4×#126

Subgroups

subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

4464 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:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC22⋊C4×#126⋊C3
maximal subgroup ofC22⋊C4×#126⋊C3, #1262
maximal normal subgroup ofC22⋊C4×#126⋊C3, #1262

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426671);