Finite Group Databasebeta

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Cayley diagram

C2×#16880 #611660

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

Aliases

direct productC2×#16880

Subgroups

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

3182 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);(9,13)(12,14)(15,16)(17,19)(18,20)(21,22)(23,24,27,26)(25,29,30,28);(3,4,5,6)(8,11)(12,14)(16,17)(20,21)(25,30)(28,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,17,19,16)(18,20,22,21)(23,25,27,30)(24,28,26,29);(7,9)(8,12)(10,13)(11,14)(15,18,19,22)(16,20,17,21)(23,26,27,24)(25,29,30,28)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#2892337
maximal normal subgroup ofC2×#2892337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 518856);