Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #150138 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | #1397 | C23 |
| prime radical | #33981 | C4 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ C4×M64 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,15,18,60,16,57,40,63)(4,21,22,79)(5,28)(6,34)(7,39)(8,9)(10,17,45,69,41,64,56,72)(11,25,42,85)(12,31)(13,37)(14,36)(19,74,73,117,65,121,91,125)(20,78,35,62,66,123,76,116)(23,52,88,59,83,113,70,122)(24,54)(26,96,86,119)(27,58,87,82)(29,103,68,111,77,61,106,48)(30,44)(32,49)(33,99)(38,53)(43,75)(46,97,115,90,100,104,118,128)(47,108,55,71,110,124,114,89)(50,120,112,92)(51,67,109,95)(80,126,107,94)(81,93,127,98)(84,101)(102,105);(2,9)(3,16)(4,22)(12,43)(13,44)(14,39)(15,57)(18,40)(19,65)(20,66)(25,85)(26,86)(27,87)(28,99)(29,77)(34,53)(35,76)(48,111)(49,102)(54,101)(60,63)(61,103)(62,116)(67,95)(68,106)(73,91)(74,121)(78,123)(92,120)(93,98)(94,126)(117,125);(1,3)(2,10)(4,23)(5,29)(6,35)(7,40)(8,16)(9,41)(11,48)(12,52)(13,55)(14,56)(15,58)(17,67)(18,36)(19,75)(20,37)(21,60)(22,83)(24,91)(25,69)(26,97)(27,88)(28,100)(30,66)(31,65)(32,106)(33,77)(34,47)(38,76)(39,45)(42,111)(43,113)(44,114)(46,99)(49,118)(50,74)(51,103)(53,110)(54,122)(57,82)(59,101)(61,109)(62,81)(63,79)(64,95)(68,105)(70,87)(71,94)(72,85)(73,84)(78,119)(80,125)(86,104)(89,126)(90,98)(92,108)(93,128)(96,123)(102,115)(107,117)(112,121)(116,127)(120,124)Polycyclic presentation:
2^10|a^2=e,d^2=fhij,e^2=g,f^2=hij,h^2=ij,i^2=j,b^a=bgj,c^a=cd,c^b=cj,d^a=df,d^b=dj,d^c=dfij,e^c=eij,e^d=ej,f^a=fh,f^c=fhj,g^c=gj,h^a=hi,h^c=hi,i^a=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72321932, #72321964, #72321979 |
| maximal normal subgroup of | #72321932, #72321964, #72321979 |
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.5);;
gap> SetPower(coll, 4, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
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);