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Cayley diagram

D8×AGL(1,ℤ/8) #6358722

order512
nilpotency class3
isoclinismisoclinic class of D8×AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productD8×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC22C2×D42
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

2379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 16 points:

(1,2,3,4,5,6,7,8);(2,8)(3,7)(4,6);(9,10)(11,13)(12,14)(15,16);(10,13)(11,16)(14,15);(10,14)(13,15)

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237138, AGL(1,ℤ/8)2, C2×D8×AGL(1,ℤ/8), …
maximal normal subgroup of#72237138, AGL(1,ℤ/8)2, C2×D8×AGL(1,ℤ/8), …
maximal quotient ofD8×#442, #102×AGL(1,ℤ/8), C2×D8×AGL(1,ℤ/8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265918);