Finite Group Databasebeta

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Cayley diagram

C2×#25741 #6443326

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

Aliases

direct productC2×#25741

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

5202 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)(7,8,12,15)(9,13,18,22)(10,19,14,11)(16,21,20,17);(3,4,6,5)(7,9,12,18)(8,13,15,22)(10,20,14,16)(11,21,19,17);(7,10,11,8,12,14,19,15)(9,16,17,13,18,20,21,22);(3,5,6,4)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6443331, C2×#6573669, C2×#6573691
maximal normal subgroup ofC2×#6443331, C2×#6573669, C2×#6573691

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350522);