Finite Group Databasebeta

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Cayley diagram

C22×#2758 #7633537

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC22×#2758

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C23 ▷ C1

6322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2);(3,4);(6,9)(7,11)(13,14,16,18)(15,19,20,17)(21,22)(23,28)(24,27)(25,26);(15,20)(17,19)(21,23,26,24)(22,27,25,28);(5,6,8,9)(7,10,11,12)(13,15,16,20)(14,17,18,19)(21,24,26,23)(22,27,25,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,22)(23,27,24,28)

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, 7540733);