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Cayley diagram

C2×C4×AΣL(1,𝔽16) #492985

order512
nilpotency class4
isoclinismisoclinic class of AΣL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×C4×AΣL(1,𝔽16)

Subgroups

subgroupG/H
centerC22×C4#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
socleC23C2×#99
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

2716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 14 points:

(1,2);(3,4,5,6);(7,8)(9,13,14,11)(10,12);(8,11)(12,13)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#495771, C2×#493995, C2×AGL(1,𝔽5)×AΣL(1,𝔽16), …
maximal normal subgroup ofC2×#495771, C2×#493995, C4×#12687, …
maximal quotient ofC2×AGL(1,𝔽5)×AΣL(1,𝔽16), C4×#12687, #72282604, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400181);