Finite Group Databasebeta

← Prev (#6443320) Next (#6443322) →

Cayley diagram

C2×#25738 #6443321

order512
nilpotency class3
isoclinismisoclinic class of #24412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#25738

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6443328
maximal normal subgroup ofC2×#6443328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350517);