Finite Group Databasebeta

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Cayley diagram

C2×#21808 #6374272

order512
nilpotency class3
isoclinismisoclinic class of #3192
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C42

Aliases

direct productC2×#21808

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6481649, C2×#6519691, C2×#6491712
maximal normal subgroup ofC2×#6481649, C2×#6519691, C2×#6491712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281468);