Finite Group Databasebeta

← Prev (#512712) Next (#512714) →

Cayley diagram

C22×#1424 #512713

order512
nilpotency class5
isoclinismisoclinic class of D32
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×#1424

Subgroups

subgroupG/H
center, socle, prime radicalC24D16
commutatorC16C23×C4
Frattini, cosocleC2×C16C24
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2);(3,4);(5,6,7,8)(10,15)(11,20)(12,21)(13,24)(16,23)(17,32)(18,35)(19,29)(22,31)(25,38)(26,27)(28,33)(30,39)(34,40)(36,37);(9,10,16,30,12,17,22,28,24,33,37,32,38,39,11,15,14,19,20,27,25,34,36,35,13,18,31,40,21,26,23,29)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#13654, #72286849, #72286857, …
maximal normal subgroup ofC22×#13654, #72286849, #72286857, …
maximal quotient of#72321409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419909);