Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #496225 |
| commutator | C22×Q8 | C23×C4 |
| Frattini, cosocle | C2×#124 | C24 |
| prime radical | #6437458 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3601 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)(4,17,19,51)(5,22,23,38)(6,26,27,11)(7,12)(9,20,33,54)(10,25)(13,29)(14,41,42,61)(15,45,24,57)(16,47,40,32)(18,49,21,52)(28,36)(30,43,59,63)(31,46)(34,53,35,56)(37,48)(39,50,44,62)(55,60,64,58);(3,13)(4,18)(5,23)(8,29)(10,36)(16,40)(17,49)(19,21)(22,38)(25,28)(30,58)(32,47)(43,55)(51,52)(59,60)(63,64);(1,3,7,13)(2,8,12,29)(4,14,18,39)(5,24,23,15)(6,16,27,40)(9,30,34,58)(10,37,36,31)(11,32,26,47)(17,50,49,41)(19,42,21,44)(20,55,53,43)(22,57,38,45)(25,48,28,46)(33,59,35,60)(51,62,52,61)(54,64,56,63);(1,4,7,18)(2,9)(3,14,13,39)(5,20,23,53)(6,19,27,21)(8,30)(10,17)(11,33)(12,34)(15,43,24,55)(16,42,40,44)(22,51)(25,56,28,54)(26,35)(29,58)(31,41)(32,59)(36,49)(37,50)(38,52)(45,61)(46,64,48,63)(47,60)(57,62)Polycyclic presentation:
2^10|a^2=e,c^2=j,d^2=gij,f^2=j,h^2=j,c^a=cg,c^b=cj,d^a=df,d^b=dgij,e^d=eh,f^a=fh,f^b=fj,f^c=fj,f^e=fij,g^d=gj,h^a=hi,h^d=hj,h^e=hj,h^f=hj,i^a=ij,i^d=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72318025, #72318072, #72318208 |
| maximal normal subgroup of | #72318025, #72318072, #72318208 |
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.10);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 6, 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.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);