Finite Group Databasebeta

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Cayley diagram

C4×#3282 #6383803

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC4×#3282

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#3282C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2876 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);(5,6)(7,8)(9,10,13,15)(11,17,20,24)(12,21,22,16)(14,18,23,19);(9,11,13,20)(10,14,15,23)(12,18,22,19)(16,24,21,17);(6,8);(10,15)(11,18)(12,22)(14,17)(19,20)(23,24)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#61066, #72284699, #72285241
maximal normal subgroup ofC4×#61066, #72284699, #72285241

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290999);