Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152018 |
| commutator | C2×#127 | C22×C4 |
| Frattini, cosocle | #3151 | C23 |
| prime radical | #6349099 | C2 |
| LCS | G ▷ C2×#127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C22×C42 ◁ #60798 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1730 conjugacy classes of subgroups — too many to draw. Only its 81 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)(3,9)(4,10)(5,21)(6,11)(7,12)(8,13)(14,30)(15,24)(16,25)(17,32)(18,23)(19,27)(20,28)(22,29)(26,31);(1,3)(2,9)(4,17,19,18,20,31,22,14)(5,8)(6,16)(7,15)(10,26,27,30,28,32,29,23)(11,25)(12,24)(13,21);(1,4,5,18,7,22,15,17,8,20,3,14,6,19,16,31)(2,10,9,23,12,29,25,32,13,28,21,30,11,27,24,26)Polycyclic presentation:
2^10|b^2=gh,c^2=d,d^2=hij,g^2=i,h^2=j,b^a=bij,c^a=ce,c^b=cf,d^a=di,d^b=dg,e^b=ej,e^c=ei,e^d=ej,f^b=fh,f^c=fgi,f^d=fhij,g^b=gi,g^c=gh,g^f=gi,h^b=hi,h^c=hj,h^f=hj,i^c=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72306540, #72306554, #72306761, … |
| maximal normal subgroup of | #72306540, #72306554, #72306761, … |
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, 2, F.7*F.8);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*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.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);