Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #21529 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C2×C4×C8 | C24 |
| socle | C2 | #210737 |
| prime radical | #33604 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ #33418 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,7,13,30,44,8,15,6,31,29,14,25,12)(3,16,19,9,21,33,54,35,17,45,20,34,48,42,50,36)(4,22,26,40,27,39,59,38,24,43,28,41,57,56,61,10)(18,49,51,64,52,63,60,55,47,32,53,62,23,37,58,46);(1,3,7,21,8,17,29,48)(2,9)(4,23,57,47,24,52,27,18)(5,20,30,50,6,19,25,54)(10,37,38,63)(11,42)(12,45)(13,35)(14,36)(15,34)(16,44)(22,55,43,46)(26,60,61,51,28,58,59,53)(31,33)(32,40,49,41)(39,62,56,64);(3,17)(4,24)(10,38)(19,20)(21,48)(22,43)(26,28)(27,57)(32,49)(37,63)(39,56)(40,41)(46,55)(50,54)(59,61)(62,64);(1,4,25,59,29,57,6,26,8,24,30,61,7,27,5,28)(2,10,31,22,13,40,12,39,15,38,11,43,14,41,44,56)(3,18,50,60,48,23,20,51,17,47,54,58,21,52,19,53)(9,32,42,62,35,37,16,46,34,49,33,64,36,63,45,55)Polycyclic presentation:
2^10|a^2=e,b^2=eghi,d^2=gi,e^2=h,g^2=i,h^2=j,i^2=j,b^a=bg,c^a=cf,c^b=chi,d^a=dgi,d^b=dij,d^c=dj,e^b=ej,e^d=ej,f^b=fj,g^a=gi,g^b=gj,i^a=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72224968, #72225065, #72225084 |
| maximal normal subgroup of | #72224968, #72225065, #72225084 |
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.5*F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);