Finite Group Databasebeta

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Cayley diagram

C2×#13528 #499523

order512
nilpotency class4
isoclinismisoclinic class of #13304
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13528

Subgroups

subgroupG/H
center, socleC24C2×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#499581, C2×#511530, C2×#509069
maximal normal subgroup ofC2×#499581, C2×#511530, C2×#509069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406719);