Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#3109 |
| commutator, Frattini, cosocle | C22×C8 | C25 |
| prime radical | #499919 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(10,15,18,29)(12,23)(13,26)(16,27,32,30)(17,33,19,28)(21,36)(22,35)(24,37)(25,38)(31,40,34,39);(1,3)(2,5)(4,7)(6,8)(9,10,12,15,13,16,23,30,14,18,24,29,26,32,37,27)(11,19,21,28,22,34,36,40,20,17,25,33,35,31,38,39);(10,16,18,32)(11,20)(12,24)(15,27,29,30)(17,34,19,31)(22,35)(23,37)(28,40,33,39);(2,6)(3,7)(9,11,14,20)(10,17,18,19)(12,25,24,21)(13,22,26,35)(15,28,29,33)(16,31,32,34)(23,38,37,36)(27,39,30,40);(3,7)(5,8)(11,20)(17,19)(21,25)(22,35)(28,33)(31,34)(36,38)(39,40)Polycyclic presentation:
2^10|a^2=h,b^2=f,c^2=h,d^2=j,f^2=i,i^2=j,b^a=bf,c^b=cij,d^a=dg,d^b=dg,d^c=dj,e^b=egh,e^d=ej,f^a=fi,f^c=fj,f^d=fj,g^b=gj,h^b=hj,i^a=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72307057 |
| maximal normal subgroup of | #72307057 |
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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
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.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);