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Cayley diagram

Q8×AGL(1,ℤ/8) #61366

order256
nilpotency class3
isoclinismisoclinic class of D4×AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productQ8×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×AGL(1,ℤ/8)C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

1005 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:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofSL(2,𝔽3)×AGL(1,ℤ/8)
maximal subgroup of#250920, #250952, #250984, …
maximal normal subgroup of#250920, #250952, #250984, …
maximal quotient ofQ8×#418, Q8×#417, C4⋊C4×AGL(1,ℤ/8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54354);