Finite Group Databasebeta

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Cayley diagram

C2×#43928 #7668674

order512
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#43928

Subgroups

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

8162 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)(9,13)(11,14)(15,16)(17,18)(19,21)(20,22);(7,8)(9,14)(10,12)(11,13)(15,17,20,19)(16,21,22,18);(3,4)(5,6)(7,8,10,12)(9,11,13,14)(15,17,20,19)(16,18,22,21);(7,8)(9,11)(10,12)(13,14)(15,18,20,21)(16,17,22,19);(7,9)(8,11)(10,13)(12,14)(15,19,20,17)(16,21,22,18)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7575870);