Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #346546 |
| commutator, Frattini, cosocle | C2×#131 | C24 |
| prime radical | #10584556 | C2 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
3369 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)(3,10)(4,19)(5,12)(6,27)(7,14)(8,15)(9,11)(13,30)(16,34)(17,43)(18,36)(20,49)(21,38)(22,53)(23,40)(24,41)(25,37)(26,42)(28,35)(29,57)(31,50)(32,33)(39,54)(44,52)(45,58)(46,63)(47,60)(48,61)(51,56)(55,62)(59,64);(1,3,8,18)(2,10,15,36)(4,7,9,32)(6,28,30,17)(11,33,19,14)(13,43,27,35)(16,45)(20,25)(21,52,31,29)(23,51)(24,48)(34,58)(37,49)(38,44,50,57)(40,56)(41,61)(46,47)(55,64)(59,62)(60,63);(1,4,8,9)(2,11,15,19)(3,7,18,32)(5,22)(6,21,30,31)(10,33,36,14)(12,39)(13,38,27,50)(16,46)(17,29,28,52)(20,51)(23,25)(24,55)(26,54)(34,59)(35,57,43,44)(37,56)(40,49)(41,60)(42,53)(45,47)(48,64)(58,62)(61,63);(1,5,7,24,9,25,18,48,8,26,32,45,4,20,3,16)(2,12,14,41,11,37,36,61,15,42,33,58,19,49,10,34)(6,23,29,46,31,54,28,55,30,51,52,64,21,22,17,47)(13,40,44,63,38,39,43,62,27,56,57,59,50,53,35,60)Polycyclic presentation:
2^10|b^2=h,c^2=h,d^2=f,e^2=j,f^2=i,i^2=j,c^a=cj,d^a=dg,d^b=df,d^c=de,e^a=eh,e^b=ej,f^b=fi,f^c=fj,g^c=ghj,g^e=gj,h^d=hj,i^b=ij67 conjugacy classes — view the full character table.
| maximal subgroup of | #72250237 |
| maximal normal subgroup of | #72250237 |
| maximal quotient of | #72256826, #72280405 |
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, 2, F.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);