Finite Group Databasebeta

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Cayley diagram

C2×#25743 #6443323

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

Aliases

direct productC2×#25743

Subgroups

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

5440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

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, 6350519);