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Cayley diagram

D4×AGL(1,ℤ/8) #61065

order256
nilpotency class3
isoclinismisoclinic class of D4×AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productD4×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

1615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 5 generators on 12 points:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#250903, #250935, #250967, …
maximal normal subgroup of#250903, #250935, #250967, …
maximal quotient ofD8×AGL(1,ℤ/8), SD16×AGL(1,ℤ/8), #6358736, …
automorphism group of#472, #473

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54053);