Finite Group Databasebeta

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Cayley diagram

C2×#22711 #6380220

order512
nilpotency class3
isoclinismisoclinic class of #3264
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#22711

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(3,4,5,6)(8,11)(12,14)(16,19)(17,21)(20,22);(7,8)(9,12)(10,11)(13,14)(16,20)(19,22);(7,9,10,13)(8,12,11,14)(15,16,21,22,18,20,17,19);(8,11)(12,14)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6384389, C2×#7540251, C2×#7548897
maximal normal subgroup ofC2×#6384389, C2×#7540251, C2×#7548897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287416);