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Cayley diagram

C2×#36028 #7629547

order512
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#36028

Subgroups

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

8166 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)(7,8,12,10)(9,14,11,13)(16,20)(21,22);(3,4,6,5)(7,9,12,11)(8,13,10,14);(3,5,6,4)(7,8,12,10)(9,14,11,13)(15,16)(17,21)(18,22)(19,20);(7,10,12,8)(9,14,11,13)(15,17,19,18)(16,21,20,22);(7,11,12,9)(8,13,10,14)(15,18,19,17)(16,22,20,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536743);