Finite Group Databasebeta

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Cayley diagram

C2×#16610 #611201

order512
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#16610

Subgroups

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

3686 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)(8,11)(9,13)(15,16,18,20)(17,21,22,19)(25,29)(28,30);(15,16)(17,19)(18,20)(21,22)(23,24,27,26)(25,30,29,28);(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,20)(19,22)(23,25,27,29)(24,28,26,30);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,17,18,22)(16,19,20,21)(23,26,27,24)(25,30,29,28)

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, 518397);