Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #12067 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #7053 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
169 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 169 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,4,3,6,9,14,12,5,7,13,10,8,15,11,16)(17,18,20,25)(19,29,31,60,33,61,34,70)(21,38)(22,42,36,73)(23,45,37,28)(24,46,48,66,49,32,50,69)(26,53)(27,56,52,44)(30,59,43,78,64,54,65,62)(35,71)(39,57,63,77,68,80,75,47)(40,74)(41,51)(55,67)(58,72)(76,79);(2,7)(3,10)(9,15)(12,16)(17,19,20,33)(18,24,25,49)(21,39,35,68)(22,43,36,65)(23,31,37,34)(26,54,51,59)(27,57,52,80)(28,48,45,50)(29,58,61,74)(30,44,64,56)(32,67,46,76)(38,70,71,60)(40,75,72,63)(41,66,53,69)(42,77,73,47)(55,78,79,62);(1,3,11,2,8,16,13,15,5,10,14,7,6,12,4,9)(19,30,31,43,33,64,34,65)(21,40,35,72)(22,36)(24,47,48,57,49,77,50,80)(26,55,51,79)(27,52)(29,59,60,78,61,54,70,62)(32,68,69,75,46,39,66,63)(38,74,71,58)(41,76,53,67)(42,73)(44,56)Polycyclic presentation:
2^10|a^2=d,b^2=j,c^2=dfhj,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,b^a=be,c^a=ch,c^b=cghij,d^b=dij,e^a=eg,e^b=egij,e^c=ei,e^d=ej,f^b=fj,g^a=gi,g^b=gij,g^c=gj,i^a=ij,i^b=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72322996, #72323007, #72323096, … |
| maximal normal subgroup of | #72322996, #72323007, #72323096, … |
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, 2, F.10);;
gap> SetPower(coll, 3, F.4*F.6*F.8*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);