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#72226883

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#136495
commutatorC4×C16C22×C4
Frattini, cosocleC2×C4×C16C23
prime radicalC2×#2246C22
LCSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ #12664 ◁ G
LECSG ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

497 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

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

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,14,7,16,27,46,9,17,34,51,8,37,36,15)(3,18,22,54,23,42,60,63,25,55,61,64,24,49,58,38)(4,10,28,12,5,31,35,47,30,39,32,48,29,50,26,13)(11,40,41,20,44,19,52,33,45,56,53,59,62,57,43,21);(2,10)(3,19)(4,26)(5,32)(6,36)(7,8)(11,41)(12,15)(13,14)(16,50)(17,39)(18,49)(20,22)(21,60)(23,40)(24,56)(25,57)(27,34)(28,29)(30,35)(31,37)(33,58)(42,55)(43,44)(45,53)(46,48)(47,51)(52,62)(54,64)(59,61);(1,3,4,20,6,22,28,19,7,23,5,33,27,60,35,56,9,25,30,59,34,61,32,57,8,24,29,21,36,58,26,40)(2,11,12,18,46,52,39,63,37,62,13,49,14,41,31,54,17,45,48,55,15,43,10,38,16,44,47,42,51,53,50,64)

Polycyclic presentation:

2^10|a^2=f,c^2=d,d^2=ghi,e^2=ghij,f^2=h,g^2=ij,h^2=j,i^2=j,b^a=be,c^a=cefghj,c^b=cegij,d^a=dg,d^b=dghj,e^a=egj,e^b=egh,e^c=ej,f^b=fhj,g^a=gi,g^b=gi,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226884, #72226889, #72226900
maximal normal subgroup of#72226884, #72226889, #72226900
maximal quotient of#72227149

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);