Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1333 |
| commutator | C2×C16 | C22×C8 |
| Frattini, cosocle | C2×#346 | C23 |
| prime radical | C2×#2246 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ C4×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,6,3,8)(9,10,11,17,13,19,12,26,28,39,27,38,14,33,23,18,16,22,25,36,32,37,31,40,15,21,24,35,29,20,30,34);(2,6)(3,4)(10,17)(11,23)(12,27)(13,29)(14,32)(15,28)(18,37)(19,34)(20,26)(21,38)(22,36)(24,31)(25,30)(33,40)(35,39);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^10|a^2=d,c^2=hij,d^2=f,e^2=ghi,f^2=ij,g^2=i,h^2=j,i^2=j,b^a=be,c^a=chij,c^b=chij,d^b=dg,e^a=eh,e^b=egh,e^d=ej,f^b=fi,g^a=gj,g^b=gij,h^b=hj,i^b=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72227292 |
| maximal normal subgroup of | #72227292 |
| maximal quotient of | #72227282 |
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.9*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);