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#72322047

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#153078
commutator, Frattini, cosocleC22×C32C23
LCSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G
LECSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

450 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 128
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 128 points:

(3,17)(4,24)(5,30)(6,26)(7,38)(8,42)(10,45)(11,50)(12,56)(13,52)(14,62)(15,66)(18,46)(19,44)(20,71)(21,81)(22,78)(23,76)(25,37)(27,87)(28,86)(29,84)(31,59)(32,70)(33,40)(34,36)(35,83)(39,91)(41,72)(43,100)(47,74)(48,110)(49,111)(51,61)(53,63)(54,118)(55,113)(57,105)(58,64)(60,117)(65,106)(67,125)(68,69)(73,123)(75,85)(77,101)(79,127)(80,128)(82,97)(88,116)(89,92)(90,98)(93,94)(95,99)(96,112)(102,108)(103,104)(107,114)(109,122)(115,120)(119,124)(121,126);(1,2)(3,18)(4,11)(5,31)(6,13)(7,14)(8,15)(9,16)(10,44)(12,36)(17,69)(19,74)(20,68)(21,73)(22,65)(23,47)(24,50)(25,57)(26,52)(27,88)(28,54)(29,55)(30,59)(32,51)(33,58)(34,56)(35,96)(37,99)(38,62)(39,63)(40,64)(41,102)(42,66)(43,67)(45,76)(46,71)(48,89)(49,104)(53,91)(60,98)(61,101)(70,107)(72,111)(75,105)(77,114)(78,121)(79,124)(80,126)(81,119)(82,109)(83,112)(84,113)(85,95)(86,118)(87,116)(90,117)(92,122)(93,120)(94,115)(97,110)(100,125)(103,108)(106,128)(123,127);(1,3)(2,10)(4,25)(5,32)(6,37)(7,41)(8,44)(9,20)(11,51)(12,57)(13,61)(14,65)(15,68)(16,47)(17,24)(18,66)(19,30)(21,26)(22,38)(23,42)(27,89)(28,92)(29,85)(31,95)(33,79)(34,76)(35,81)(36,77)(39,101)(40,75)(43,104)(45,50)(46,56)(48,52)(49,62)(53,73)(54,119)(55,114)(58,109)(59,69)(60,122)(63,105)(64,107)(67,126)(70,87)(71,84)(72,86)(74,113)(78,83)(80,100)(82,91)(88,124)(90,128)(93,97)(94,103)(96,110)(98,127)(99,116)(102,125)(106,118)(108,117)(111,112)(115,121)(120,123)

Polycyclic presentation:

2^10|d^2=gh,e^2=ghj,g^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=cd,c^b=cdegj,d^a=dg,d^c=dg,e^a=egj,e^b=ej,e^c=egj,f^c=fj,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72322063, #72322916, #72322965
maximal normal subgroup of#72322063, #72322916, #72322965

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