Finite Group Databasebeta

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Cayley diagram

C2×#15676 #609313

order512
nilpotency class2
isoclinismisoclinic class of #15257
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#15676

Subgroups

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

3802 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)(8,10)(11,12)(13,16)(14,15)(17,18)(19,20,22,24)(21,23,25,26);(12,14)(16,17)(19,21,22,25)(20,23,24,26);(7,8,9,10)(11,13,15,18)(12,16,14,17)(19,21)(20,23)(22,25)(24,26);(3,4)(5,6)(11,14,15,12)(13,17,18,16)(19,20)(21,23)(22,24)(25,26)

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, 516509);