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Cayley diagram

C2×#15974 #609910

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

Aliases

direct productC2×#15974

Subgroups

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

3714 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);(3,4,5,6)(11,12,15,13)(14,17,18,16)(20,23)(21,25);(8,10)(12,13)(14,18)(19,20,22,23)(21,26,25,24);(7,8)(9,10)(11,13,15,12)(14,17,18,16)(19,20)(21,24)(22,23)(25,26);(3,4)(5,6)(11,14,15,18)(12,16,13,17)(19,21,22,25)(20,24,23,26)

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, 517106);