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Cayley diagram

C2×#15008 #608060

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#15008

Subgroups

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

4054 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);(4,5)(11,12,15,16)(13,18,17,14)(21,25)(23,26);(8,10)(12,16)(13,17)(19,20,22,24)(21,23,25,26);(3,4,6,5)(7,8)(9,10)(11,13,15,17)(12,14,16,18)(19,20)(21,23)(22,24)(25,26);(3,5,6,4)(11,14)(12,17)(13,16)(15,18)(19,21)(20,23)(22,25)(24,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515256);