Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152018 |
| commutator | C2×#127 | C22×C4 |
| Frattini, cosocle | #3151 | C23 |
| prime radical | #60801 | C22 |
| LCS | G ▷ C2×#127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C22×C42 ◁ #60801 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1513 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1513 conjugacy classes of subgroups — too many to draw. Only its 81 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)(3,13)(4,19)(5,11)(6,12)(7,10)(8,15)(9,22)(14,29)(16,30)(17,31)(18,32)(20,38)(21,33)(23,34)(24,49)(25,48)(26,36)(27,53)(28,37)(35,40)(39,44)(41,61)(42,55)(43,56)(45,59)(46,62)(47,57)(50,54)(51,60)(52,64)(58,63);(1,3,5,17,6,18,28,47,7,15,26,44,21,16,27,45)(2,8,11,31,12,30,37,57,10,13,36,39,33,32,53,59)(4,20,24,50,25,43,52,46,22,14,40,41,23,42,51,63)(9,29,35,54,34,55,60,62,19,38,49,61,48,56,64,58);(1,4,21,23,7,22,6,25)(2,9,33,48,10,19,12,34)(3,14,15,20)(5,24,27,51,26,40,28,52)(8,29,13,38)(11,35,53,64,36,49,37,60)(16,42,18,43)(17,46,44,63)(30,56,32,55)(31,58,39,62)(41,47,50,45)(54,57,61,59)Polycyclic presentation:
2^10|b^2=f,c^2=ghj,f^2=hi,g^2=i,h^2=j,b^a=bd,c^a=cj,c^b=ce,d^b=di,d^c=dj,e^a=ej,e^b=eg,e^c=eh,f^a=fi,f^c=fgi,f^d=fj,f^e=fhij,g^b=gh,g^c=gi,g^e=gi,h^b=hj,h^c=hi,h^e=hj,i^b=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72306636, #72306732, #72306927, … |
| maximal normal subgroup of | #72306636, #72306732, #72306927, … |
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.6);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
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.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);