Finite Group Databasebeta

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Cayley diagram

C2×C4×#360 #493064

order512
nilpotency class4
isoclinismisoclinic class of #360
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×C4×#360

Subgroups

subgroupG/H
centerC22×C4#102
commutatorC8C22×C42
Frattini, cosocleC22×C8C24
socleC23C2×#102
prime radicalC22×#135C22
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#495527, C2×#494991, …
maximal normal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#495527, C2×#494991, …
maximal quotient ofC2×C4×#1333, C2×C8×#360, C2×AGL(1,𝔽5)×#360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400260);