Finite Group Databasebeta

← Prev (#72322044) Next (#72322046) →

#72322045

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#153264
commutator, Frattini, cosocleC22×C32C23
prime radical#513054C2
LCSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ #13549 ◁ G
LECSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

391 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 85 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fi,d^2=gh,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=cd,c^b=cdeg,d^a=dg,d^b=dj,d^c=dgj,e^a=eg,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, #72322926, #72322960
maximal normal subgroup of#72322063, #72322926, #72322960

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