Finite Group Databasebeta

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Cayley diagram

C2×#51782 #7707769

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#51782

Subgroups

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

8248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7614965);