Finite Group Databasebeta

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Cayley diagram

C2×#15388 #608922

order512
nilpotency class2
isoclinismisoclinic class of #15257
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#15388

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

3682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 26 points:

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

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, 516118);