Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8535 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C22×#120 | C23 |
| prime radical | #15947 | C22 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
2020 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 2020 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(2,8)(9,23)(10,24)(11,18)(12,28)(25,26)(27,32)(29,30);(1,2,5,8,6,11,4,18)(3,12,15,29,16,9,19,25)(7,10,21,24,22,27,20,32)(13,26,14,23,17,30,31,28);(1,3,7,17)(2,9,11,12)(4,19,20,14)(5,15,21,31)(6,16,22,13)(8,23,18,28)(10,26,27,30)(24,25,32,29)Polycyclic presentation:
2^10|b^2=f,c^2=ghj,f^2=i,b^a=be,c^b=cd,d^a=dg,d^b=dg,d^c=dij,e^b=eh,e^c=egj,e^d=ej,f^a=fh,f^c=fg,f^d=fj,f^e=fi,g^b=gj,h^b=hi52 conjugacy classes — view the full character table.
| maximal subgroup of | #72263143, #72263149, #72263171, … |
| maximal normal subgroup of | #72263143, #72263149, #72263171, … |
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.6);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
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.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);