Finite Group Databasebeta

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Cayley diagram

C2×#35520 #7628898

order512
nilpotency class2
isoclinismisoclinic class of #35346
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#35520

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

8936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7631302, C2×#7631308, C2×#7628905, …
maximal normal subgroup ofC2×#7631302, C2×#7631308, C2×#7628905, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536094);