Finite Group Databasebeta

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Cayley diagram

C2×#16785 #611524

order512
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#16785

Subgroups

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

3030 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)(7,8)(9,11)(10,13)(12,14)(15,16)(17,21)(18,20)(19,22)(23,24,26,28)(25,27,29,30);(8,9)(12,13)(15,16,18,20)(17,19,22,21)(23,25,26,29)(24,27,28,30);(3,4,6,5)(7,9,11,8)(10,12,14,13)(15,16)(17,21)(18,20)(19,22)(23,25)(24,27)(26,29)(28,30);(3,5,6,4)(7,10,11,14)(8,12,9,13)(15,17)(16,19)(18,22)(20,21)(23,24)(25,27)(26,28)(29,30)

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, 518720);