Finite Group Databasebeta

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Cayley diagram

C2×#47874 #7703944

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

Aliases

direct productC2×#47874

Subgroups

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

8574 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);(3,4,5,6)(15,16)(17,20)(18,19)(21,22);(7,8)(9,13)(10,14)(11,12)(16,18)(20,21);(3,4)(5,6)(8,12)(9,14)(15,16,19,18)(17,21,22,20);(7,9)(8,10)(11,14)(12,13)(15,17,19,22)(16,20,18,21);(7,10)(8,9)(11,13)(12,14)(15,18,19,16)(17,21,22,20)

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, 7611140);