Finite Group Databasebeta

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Cayley diagram

C2×#59409 #7665872

order512
nilpotency class2
isoclinismisoclinic class of #58990
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#59409

Subgroups

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

9956 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);(4,5)(8,11)(9,13);(4,5)(7,8,10,11)(9,12,13,14)(17,21)(19,22);(3,4,6,5)(7,8)(9,12)(10,11)(13,14)(15,16,18,20)(17,19,21,22);(3,5,6,4)(7,8)(9,12)(10,11)(13,14)(15,17)(16,19)(18,21)(20,22);(7,9)(8,12)(10,13)(11,14)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72261501, #72261516, C2×#7699554, …
maximal normal subgroup of#72261501, #72261516, C2×#7699554, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7573068);