Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1368 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | #1311 | C23 |
| prime radical | C23 | #1368 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ C2×#1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 512 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,3,7,5,8,10,14,4,6,9,12,11,13,15,16)(17,18)(19,31,33,58,35,59,20,37)(21,41,42,74)(22,44)(23,47)(24,30)(25,34,26,39,50,65,53,62)(27,43,55,75)(28,45)(29,48)(32,61,38,69)(36,68,64,60,67,79,40,73)(46,57)(49,63,52,71)(51,77,54,66,76,72,78,80)(56,70);(2,6)(7,12)(8,13)(14,16)(17,19,21,34,22,36,43,66,23,33,32,62,45,64,63,77,24,35,42,65,46,67,75,80,48,20,38,39,70,40,71,72)(18,25,27,31,28,51,41,60,29,26,52,37,44,54,69,68,30,50,55,59,56,76,74,73,47,53,49,58,57,78,61,79);(17,20,21,39,22,40,43,72,23,19,32,34,45,36,63,66,24,33,42,62,46,64,75,77,48,35,38,65,70,67,71,80)(18,26,27,37,28,54,41,68,29,50,52,59,44,76,69,73,30,53,55,58,56,78,74,79,47,25,49,31,57,51,61,60)Polycyclic presentation:
2^10|a^2=d,b^2=eh,c^2=eh,d^2=f,e^2=g,f^2=h,g^2=i,i^2=j,b^a=be,c^a=ceh,c^b=cj,d^b=dij,d^c=dij,e^a=eg,e^d=ej,f^b=fj,f^c=fj,g^a=gi,i^a=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72231933, #72231943, #72231988, … |
| maximal normal subgroup of | #72231933, #72231943, #72231988, … |
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.5*F.8);;
gap> SetPower(coll, 3, F.5*F.8);;
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.5*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);