Finite Group Databasebeta

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Cayley diagram

C22×#2586 #7628001

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC22×#2586

Subgroups

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

7642 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)(10,12)(13,14,16,17)(15,19,20,18)(23,27)(26,28);(14,17)(18,19)(21,22,25,24)(23,28,27,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,27)(22,26,24,28);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,18,17,19)(21,24,25,22)(23,28,27,26)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535197);