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Cayley diagram

C2×#52710 #8092108

order512
nilpotency class2
isoclinismisoclinic class of #52308
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#52710

Subgroups

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

5758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

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

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

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, 7999304);