Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #135908 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| prime radical | C2×#508 | C2×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 | |
382 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 382 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:
(1,2,5,13,8,11,7,33)(3,15,18,47,19,48,17,49)(4,21,26,56,24,58,28,41)(6,30,29,43,32,10,25,14)(9,34,37,50,38,51,36,16)(12,23,45,62,42,61,57,27)(20,54,31,44,53,63,52,35)(22,46,59,39,60,64,55,40);(1,3)(2,9)(4,22)(5,18)(6,31)(7,17)(8,19)(10,40)(11,38)(12,44)(13,37)(14,46)(15,21)(16,27)(20,25)(23,34)(24,60)(26,59)(28,55)(29,53)(30,39)(32,52)(33,36)(35,42)(41,49)(43,64)(45,63)(47,56)(48,58)(50,62)(51,61)(54,57);(2,10,11,30)(3,16,17,34,19,50,18,51)(4,23,24,61)(5,7)(6,29)(9,35,36,54,38,44,37,63)(12,45)(13,43,33,14)(15,46,47,40,48,64,49,39)(20,55,52,22,53,59,31,60)(25,32)(26,27,28,62)(41,56)(42,57)Polycyclic presentation:
2^10|a^2=f,c^2=d,d^2=g,e^2=hi,f^2=j,h^2=ij,i^2=j,b^a=be,c^a=cfhi,c^b=ceh,d^a=dj,d^b=di,e^a=eh,e^b=eh,e^c=ehij,e^d=ej,f^c=fj,g^b=gj,h^a=hi,h^b=hi,h^c=hij,i^a=ij,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72226979, #72227146, #72227147, … |
| maximal normal subgroup of | #72226979, #72227146, #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.6);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.9*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.6*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, 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);;
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);