Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #1333 |
| commutator | C16 | C22×C16 |
| Frattini, cosocle | #1311 | C23 |
| socle | C22 | C2×#1333 |
| prime radical | C2×#508 | C8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,4,8,5,9,12,15,6,10,13,16,3,7,11,14)(17,18)(19,30,32,59,34,60,31,58)(20,26)(21,38,40,71)(22,43)(23,47)(24,29)(25,33,48,62,49,66,51,63)(27,41,54,74)(28,45)(35,69,64,61,68,53,44,78)(36,70,42,57)(37,56)(39,55,72,50,73,80,67,76)(46,77)(52,65,79,75);(17,19)(18,25)(20,32)(21,39)(22,44)(23,31)(24,34)(26,51)(27,53)(28,55)(29,49)(30,57)(33,65)(35,37)(36,67)(38,59)(40,73)(41,62)(42,72)(43,76)(45,68)(46,64)(47,48)(50,77)(52,78)(54,69)(56,80)(58,71)(60,70)(61,79)(63,74)(66,75);(1,3,6,5)(2,7,10,9)(4,11,13,12)(8,14,16,15)(17,20,24,23)(18,26,29,47)(19,31,34,32)(21,36,40,42)(22,37,46,45)(25,48,49,51)(27,52,54,79)(28,43,56,77)(30,58,60,59)(33,62,66,63)(35,44,68,64)(38,70,71,57)(39,72,73,67)(41,65,74,75)(50,80,76,55)(53,61,69,78)Polycyclic presentation:
2^10|a^2=d,c^2=hj,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,b^a=be,c^b=cj,d^b=dij,e^a=eg,e^b=egij,e^d=ej,f^b=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij136 conjugacy classes — view the full character table.
| maximal subgroup of | #72322405, #72322417, #72322430, … |
| maximal normal subgroup of | #72322405, #72322417, #72322430, … |
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.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, 2, F.3*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, 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, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);