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AGL(1,𝔽256) #72223008

order65280
automorphism?
homologyAbelianization (H1) = C3×C5×C17 · Schur multiplier (H2) = C1

Aliases

affine general linearAGL(1,𝔽256)

Splittings

Fitting, socle, derived, SylowC28 ⋊ C3×C5×C17
Hall#72223007 ⋊ C17
Hall#72222992 ⋊ C5
Hall#72222967 ⋊ C3
HallC28⋊C17 ⋊ C3×C5
HallC24⋊(C24⋊C5) ⋊ C3×C17
HallC22⋊(C22⋊(C22⋊A4)) ⋊ C5×C17

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC28C3×C5×C17
LCSG ▷ C28
UCSC1

1666 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has 1666 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

3,5,17,2^8|d^a=e,d^b=efg,d^c=eh,e^a=de,e^b=deg,e^c=dhi,f^a=fg,f^b=ef,f^c=dfghj,g^a=f,g^b=dg,g^c=dfgjk,h^a=ehi,h^b=egik,h^c=dhk,i^a=deh,i^b=egj,i^c=defgij,j^a=gik,j^b=defjk,j^c=deghij,k^a=defghijk,k^b=dgh,k^c=efgi

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [3, 5, 17, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.4*F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 2, F.4*F.7);;
gap> SetConjugate(coll, 7, 3, F.4*F.6*F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.5*F.7*F.9*F.11);;
gap> SetConjugate(coll, 8, 3, F.4*F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.4*F.5*F.8);;
gap> SetConjugate(coll, 9, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.4*F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.7*F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.4*F.5*F.6*F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.4*F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 11, 1, F.4*F.5*F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 11, 3, F.5*F.6*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);