Finite Group Databasebeta

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Cayley diagram

C2×#36051 #7629487

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

Aliases

direct productC2×#36051

Subgroups

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

8366 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)(7,8,13,14)(9,10,12,11)(15,16,19,22)(17,20,18,21);(3,4,6,5)(7,9,13,12)(8,11,14,10)(15,17,19,18)(16,20,22,21);(3,5,6,4)(7,10,13,11)(8,9,14,12)(15,16)(17,20)(18,21)(19,22);(7,11,13,10)(8,9,14,12)(15,18,19,17)(16,21,22,20);(7,12,13,9)(8,11,14,10)

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, 7536683);