Finite Group Databasebeta

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Cayley diagram

C4×#2395 #514036

order512
nilpotency class2
isoclinismisoclinic class of #392
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC4×#2395

Subgroups

subgroupG/H
centerC22×C42C23
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C42 ◁ G
LECSG ▷ C24 ▷ C1

3798 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);(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(5,6)(7,8)(10,13)(14,16)(17,19,20,23)(18,21,22,24);(6,8)(18,22)(21,24);(9,11,12,15)(10,14,13,16)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72281212, #72281442
maximal normal subgroup of#72281212, #72281442

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421232);