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Cayley diagram

Q16×AGL(1,ℤ/8) #6358764

order512
nilpotency class3
isoclinismisoclinic class of D8×AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productQ16×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC22C2×D42
commutator, Frattini, cosocleC42C25
prime radicalC2×AGL(1,ℤ/8)D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16);(17,18)(19,21)(20,22)(23,24);(18,21)(19,24)(22,23);(18,22)(21,23)

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#135×AGL(1,ℤ/8), SD32×AGL(1,ℤ/8)
maximal normal subgroup of#135×AGL(1,ℤ/8), SD32×AGL(1,ℤ/8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265960);