Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#1368 |
| commutator | C16 | C23×C8 |
| Frattini, cosocle | #346 | C24 |
| prime radical | C2×#41 | #106 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C4×#131 ◁ #33326 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,4,8,5,12,17,26,6,10,19,27,20,32,42,34,7,11,18,25,21,33,41,52,22,31,43,24,45,50,44,51)(3,9,14,23,15,30,38,48,16,29,39,49,40,55,60,56,13,28,35,46,36,54,57,62,37,53,58,47,59,63,61,64);(2,8,10,24,11,25,31,27)(4,17,18,41)(5,20)(6,22)(9,23,29,47,28,46,53,49)(12,34,32,26,33,51,50,52)(14,38,35,57)(15,40)(16,37)(19,44,43,42)(21,45)(30,56,55,48,54,64,63,62)(36,59)(39,61,58,60);(1,3)(2,9)(4,14)(5,15)(6,16)(7,13)(8,23)(10,29)(11,28)(12,30)(17,38)(18,35)(19,39)(20,40)(21,36)(22,37)(24,47)(25,46)(26,48)(27,49)(31,53)(32,55)(33,54)(34,56)(41,57)(42,60)(43,58)(44,61)(45,59)(50,63)(51,64)(52,62);(3,13)(9,28)(14,35)(15,36)(16,37)(23,46)(29,53)(30,54)(38,57)(39,58)(40,59)(47,49)(48,62)(55,63)(56,64)(60,61)Polycyclic presentation:
2^10|a^2=f,b^2=e,e^2=g,f^2=h,h^2=i,i^2=j,b^a=bfhij,d^c=dj,e^a=eij,f^b=fh,f^e=fj,g^a=gj,h^b=hi,i^b=ij130 conjugacy classes — view the full character table.
| maximal subgroup of | #72224981, #72224982, #72225087, … |
| maximal normal subgroup of | #72224981, #72224982, #72225087, … |
| maximal quotient of | #72231930, #72231933, #72231940, … |
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, 2, F.5);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);