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Cayley diagram

C2×#15139 #608372

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#15139

Subgroups

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

3750 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)(11,12,16,14)(13,18,15,17)(21,26)(22,24);(8,10)(11,13,16,15)(12,17,14,18)(19,20,23,25)(21,24,26,22);(7,8)(9,10)(11,14,16,12)(13,18,15,17)(19,21)(20,24)(22,25)(23,26);(3,4,5,6)(11,15,16,13)(12,18,14,17)(19,22,23,24)(20,21,25,26)

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, 515568);