Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13424 |
| commutator, Frattini, cosocle | C2×#127 | C24 |
| prime radical | C22×#453 | C22 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,12,21)(4,14)(5,8)(6,10)(9,16)(11,25,29,31)(13,23,19,28)(15,22,30,32)(17,27)(18,24)(20,26);(1,3,5,12)(2,7)(4,15,17,11,16,30,20,29)(6,19,18,13)(8,21)(9,25,26,32,14,31,27,22)(10,28)(23,24);(2,8)(7,21)(9,14)(10,24)(22,32)(23,28)(25,31)(26,27);(1,4,5,16)(2,9,8,14)(3,11,12,29)(6,20,18,17)(7,22,21,32)(10,26,24,27)(13,15,19,30)(23,25,28,31)Polycyclic presentation:
2^10|a^2=e,b^2=gij,d^2=j,g^2=i,h^2=j,b^a=begi,c^a=cj,d^a=df,d^b=defj,e^b=ej,f^b=fh,g^a=gij,g^d=ghj,g^f=gi,h^a=hi,h^b=hij,h^d=hj,h^f=hj,i^d=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72306886, #72306908, #72306935, … |
| maximal normal subgroup of | #72306886, #72306908, #72306935, … |
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.5);;
gap> SetPower(coll, 2, F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);