Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #365607 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ #10711 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,11,31)(3,13,16,45)(5,23)(6,27)(10,30,38,39)(12,41,40,32)(14,47)(15,50,52,44)(17,24,28,46)(18,53,54,61)(19,55,57,58)(20,59)(22,60)(25,36)(26,43)(29,49)(33,63)(35,42)(48,62)(51,56);(1,2,5,8,6,12,23,32,7,11,26,31,27,40,43,41)(3,14,13,35,17,47,46,34,16,49,45,42,28,29,24,9)(4,18,20,53,22,38,59,30,21,54,25,61,60,10,36,39)(15,51,50,33,19,56,58,37,52,62,44,63,57,48,55,64);(1,3)(2,9)(4,19)(5,24)(6,28)(7,16)(8,29)(10,37)(11,34)(12,42)(13,43)(14,41)(15,22)(17,27)(18,33)(20,50)(21,57)(23,45)(25,44)(26,46)(30,48)(31,47)(32,49)(35,40)(36,58)(38,64)(39,56)(51,53)(52,60)(54,63)(55,59)(61,62);(1,4,7,21)(2,10)(3,15)(5,25,26,20)(6,22,27,60)(8,30)(9,33,34,63)(11,38)(12,18)(13,44)(14,48,49,56)(16,52)(17,19)(23,36,43,59)(24,58)(28,57)(29,62,47,51)(31,39)(32,61)(35,37,42,64)(40,54)(41,53)(45,50)(46,55)Polycyclic presentation:
2^10|a^2=e,b^2=f,d^2=ej,f^2=i,i^2=j,b^a=bf,c^a=cfg,c^b=cf,d^a=dh,d^b=di,d^c=di,e^b=ej,e^c=ej,f^a=fi,f^c=fij,f^d=fj,g^d=gj,h^c=hj,i^a=ij,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72320733, #72320921, #72343946 |
| maximal normal subgroup of | #72320733, #72320921, #72343946 |
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, 2, F.6);;
gap> SetPower(coll, 4, F.5*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
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);