Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139624 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #2308 | C23 |
| prime radical | #251854 | C2 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 1406 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9)(3,12,13,28,15,8,16,25)(5,10,20,26)(6,11,18,27)(7,17,23,21)(14,24,30,32,19,22,29,31);(2,7,11,10)(8,24,12,22)(9,26,27,23)(14,19)(25,31,28,32)(29,30);(1,3)(2,8,11,12)(4,16,18,13)(5,19,21,14)(6,15)(7,22)(9,25)(10,24)(17,30)(20,29)(23,32,26,31)(27,28)Polycyclic presentation:
2^10|a^2=d,b^2=i,c^2=ghj,d^2=g,e^2=gj,b^a=be,c^a=cf,c^b=cf,d^b=di,d^c=dh,e^a=egij,e^b=eg,e^c=eh,e^d=ej,f^a=fh,f^c=fi,f^d=fi,f^e=fij,g^c=gj,h^a=hi,h^b=hi,h^c=hj,h^d=hj,h^e=hj,i^a=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72284595, #72284615, #72284859, … |
| maximal normal subgroup of | #72284595, #72284615, #72284859, … |
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.9);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);