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Cayley diagram

C2×#16686 #611352

order512
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C42

Aliases

direct productC2×#16686

Subgroups

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

3534 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);(4,6)(7,8,10,12)(9,13,14,11)(17,21)(19,22)(23,24)(25,29)(26,28)(27,30);(15,16,18,20)(17,19,21,22)(23,24,26,28)(25,27,30,29);(7,9,10,14)(8,11,12,13)(15,16)(17,19)(18,20)(21,22)(23,24)(25,29)(26,28)(27,30);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,17,18,21)(16,19,20,22)(23,25)(24,27)(26,30)(28,29)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 518548);