Finite Group Databasebeta

← Prev (#514912) Next (#514914) →

Cayley diagram

C4×#2557 #514913

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC4×#2557

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

3648 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)(11,15)(12,14)(13,16)(17,18)(19,23)(20,21)(22,24);(5,6,7,8)(9,11,12,16)(10,13,14,15)(18,21)(22,23);(6,8)(18,21)(22,23);(10,14)(13,15)(17,19,20,24)(18,22,21,23)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303655, #72303657, C4×#35695
maximal normal subgroup of#72303655, #72303657, C4×#35695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422109);