Finite Group Databasebeta

← Prev (#358274) Next (#358276) →

Cayley diagram

C2×#11263 #358275

order512
nilpotency class3
isoclinismisoclinic class of #2031
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11263

Subgroups

subgroupG/H
centerC22×C4#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23#393
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(7,8)(9,16)(10,18,20,17)(11,14)(12,15)(13,19,21,22);(3,4)(5,6)(7,9,11,8,12,14,16,15)(10,18,19,21,20,17,22,13);(4,6)(7,10,11,19,12,20,16,22)(8,13,9,17,15,21,14,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235150, #72235493
maximal normal subgroup of#72235150, #72235493
maximal quotient ofC2×#123745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265471);