Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139908 |
| commutator | C23×Q8 | C22×C4 |
| Frattini, cosocle | #2310 | C23 |
| prime radical | #417567 | C2 |
| LCS | G ▷ C23×Q8 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35327 ◁ G | |
| LECS | G ▷ #2310 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1423 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1423 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,10)(3,13,15,41,17,43,47,58)(5,11,18,34)(6,23,19,35)(7,8,20,27)(9,29,31,49,26,53,33,16)(12,37,38,25)(14,40,44,62,46,42,64,63)(21,28,50,54)(22,39,51,60)(24,36,52,61)(30,55,56,45,32,59,57,48);(2,8)(3,14)(5,21)(9,30)(12,23)(13,40)(15,44)(16,48)(17,46)(18,50)(22,24)(26,32)(29,55)(31,57)(33,56)(34,54)(41,63)(42,43)(45,49)(47,64)(51,52)(53,59)(58,62)(60,61);(1,3,5,14,7,17,21,46)(2,9,11,32,8,26,28,30)(4,15,18,44,20,47,50,64)(6,16,24,48,25,49,22,45)(10,33,34,56,27,31,54,57)(12,13,39,42,23,43,36,40)(19,29,52,55,37,53,51,59)(35,41,61,63,38,58,60,62)Polycyclic presentation:
2^10|a^2=d,c^2=e,d^2=h,e^2=j,f^2=j,b^a=bghj,c^a=cf,c^b=ce,d^b=di,d^c=dg,e^a=eh,e^b=ej,f^a=fgj,f^b=fh,f^c=fhj,f^d=fi,f^e=fj,g^a=gi,g^d=gj,h^c=hj,i^a=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72284297 |
| maximal normal subgroup of | #72284297 |
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.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);