Finite Group Databasebeta

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Cayley diagram

C4×#2436 #514835

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4

Aliases

direct productC4×#2436

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

5758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#35250
maximal normal subgroup ofC4×#35250

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422031);