Finite Group Databasebeta

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Cayley diagram

C2×#56543 #8099237

order512
nilpotency class2
isoclinismisoclinic class of #53390
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#56543

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

5908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8006433);