Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #136707 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| prime radical | C2×#2263 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ #11405 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
477 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,11,6,10,8,32)(3,14)(5,22,25,44,27,61,20,47)(7,29,21,40,26,62,31,13)(9,34,36,56)(12,43,45,28,39,24,42,60)(15,48,23,46)(16,41,54,38)(17,52)(18,35)(19,58)(30,63)(33,51)(37,59,49,53)(50,57)(55,64);(1,3)(2,9)(4,19)(5,23)(6,17)(7,30)(8,33)(10,36)(11,41)(12,35)(13,46)(14,47)(15,27)(16,24)(18,21)(20,59)(22,51)(25,53)(26,57)(28,34)(29,49)(31,55)(32,38)(37,62)(39,64)(40,48)(42,50)(43,54)(44,52)(45,63)(56,60)(58,61);(2,10)(3,15)(4,8)(5,24)(7,21)(9,35)(12,44)(13,29)(14,48)(16,30)(17,23)(18,56)(19,59)(20,60)(22,45)(25,28)(26,31)(27,43)(33,53)(34,55)(36,64)(37,51)(38,50)(39,47)(40,62)(41,63)(42,61)(46,52)(49,58)(54,57)Polycyclic presentation:
2^10|a^2=d,d^2=g,e^2=hj,h^2=i,i^2=j,b^a=be,c^a=cf,c^b=cej,d^b=di,d^c=dj,e^a=eh,e^b=ehi,e^c=ehi,e^d=ej,f^a=fj,f^b=fj,g^b=gj,h^a=hi,h^b=hij,h^c=hij,i^a=ij,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72227072, #72227141, #72227147, … |
| maximal normal subgroup of | #72227072, #72227141, #72227147, … |
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.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);