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Cayley diagram

C2×#35607 #7628994

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

Aliases

direct productC2×#35607

Subgroups

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

9300 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)(7,8)(9,14)(10,12)(11,13)(15,16,18,20)(17,19,21,22);(3,4,5,6)(7,9,11,10)(8,12,13,14);(9,10)(12,14)(15,16)(17,19)(18,20)(21,22);(7,10,11,9)(8,12,13,14);(7,8)(9,14)(10,12)(11,13)(15,17,18,21)(16,19,20,22)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7672086
maximal normal subgroup ofC2×#7672086

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536190);