Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13082 |
| commutator | C22×Q8 | C23×C4 |
| Frattini, cosocle | C2×#124 | C24 |
| prime radical | #6437453 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3966 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)(3,8)(4,17,19,52,22,53,21,40)(5,10)(6,24,18,41)(7,12)(9,20,35,54,34,56,36,26)(11,27,37,28)(13,42,44,64,43,63,46,61)(14,32)(15,48,25,57)(16,31)(23,38)(29,45,59,58,33,55,60,49)(30,50,39,51)(47,62);(1,3)(2,8)(4,13)(5,14)(6,25)(7,16)(9,33)(10,32)(11,39)(12,31)(15,18)(17,42)(19,44)(20,55)(21,46)(22,43)(23,47)(24,57)(26,58)(27,51)(28,50)(29,34)(30,37)(35,60)(36,59)(38,62)(40,61)(41,48)(45,56)(49,54)(52,64)(53,63);(1,4,18,54)(2,9,11,40)(3,13,25,58)(5,22,6,26)(7,19,27,56)(8,29,30,61)(10,34,37,52)(12,35,41,17)(14,43,15,49)(16,44,50,55)(20,23,21,28)(24,53,38,36)(31,59,57,42)(32,33,39,64)(45,47,46,51)(48,63,62,60);(2,10)(3,14)(6,18)(9,34)(12,38)(13,43)(16,47)(20,56)(26,54)(27,28)(30,39)(35,36)(42,63)(44,46)(48,57)(61,64)Polycyclic presentation:
2^10|a^2=e,c^2=efg,e^2=i,g^2=j,h^2=j,c^a=cg,c^b=cf,d^a=dij,d^b=dj,d^c=dij,e^c=eh,f^a=fj,f^c=fj,g^a=gh,g^b=gj,g^c=ghj,g^d=gj,g^e=gij,h^a=hi,h^c=hi,h^g=hj,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72306296, #72317799, #72317856, … |
| maximal normal subgroup of | #72306296, #72317799, #72317856, … |
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, 3, F.5*F.6*F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);