Finite Group Databasebeta

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Cayley diagram

C4×#2377 #513770

order512
nilpotency class2
isoclinismisoclinic class of #560
automorphism?
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C26×C42

Aliases

direct productC4×#2377

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC22C23×C42
Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

3710 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)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24);(9,10)(11,16)(12,13)(14,15)(17,19)(18,21)(20,23)(22,24);(10,13)(14,16)(18,22)(19,23);(9,11,12,15)(10,14,13,16)(18,22)(21,24)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72304039, #72304094, #72304125
maximal normal subgroup of#72304039, #72304094, #72304125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420966);