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Cayley diagram

C2×#15049 #608133

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#15049

Subgroups

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

3230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(4,5)(8,11)(12,14)(15,16,18,20)(17,19,21,22)(23,24,27,28)(25,26,30,29);(3,4,6,5)(8,11)(12,14)(24,28)(25,30);(3,4,6,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22)(23,25,27,30)(24,29,28,26);(3,5,6,4)(7,9,10,13)(8,12,11,14)(15,17,18,21)(16,19,20,22)(23,26,27,29)(24,30,28,25)

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, 515329);