Finite Group Databasebeta

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Cayley diagram

C2×#15438 #608976

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

Aliases

direct productC2×#15438

Subgroups

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

3986 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)(11,12,15,14)(13,17,18,16)(20,21)(22,26);(8,10)(13,18)(16,17)(19,20,23,21)(22,25,26,24);(7,8)(9,10)(11,13,15,18)(12,16,14,17)(19,21,23,20)(22,25,26,24);(3,4)(5,6)(11,14,15,12)(13,17,18,16)(19,22)(20,24)(21,25)(23,26)

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, 516172);