Finite Group Databasebeta

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Cayley diagram

C2×C4×#509 #7622600

order512
nilpotency class4
isoclinismisoclinic class of #509
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×C4×#509

Subgroups

subgroupG/H
centerC22×C4C2×D8
commutatorC8C24×C4
Frattini, cosocleC2×C8C25
socleC23C22×D8
prime radicalC22×#509C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

2204 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)(9,12)(10,19)(11,13)(14,16)(15,17)(18,20)(21,22);(8,12)(9,16)(10,17)(13,20)(14,15)(18,22)(19,21);(8,13)(12,20)(14,21)(15,19)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#7623096, C2×#494991, …
maximal normal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#7623096, C2×#494991, …
maximal quotient ofC2×C4×#2177, C2×C8×#509, C2×AGL(1,𝔽5)×#509, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529796);