Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C22×C8 | C25 |
| prime radical | #7623308 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ #20384 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
2857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,29)(20,30)(21,31)(22,26)(23,27)(24,32)(25,28)(33,39)(34,37)(35,40)(36,38);(1,3)(2,5)(4,7)(6,8)(11,19,22,34)(12,23)(16,29,26,37)(17,27)(20,33,24,35)(21,36)(30,39,32,40)(31,38);(2,6)(3,7)(11,20,22,24)(12,21)(13,25)(16,30,26,32)(17,31)(18,28)(19,33,34,35)(23,36)(27,38)(29,39,37,40);(3,7)(5,8)(10,15)(16,26)(17,27)(18,28)(29,37)(30,32)(31,38)(39,40);(9,11,21,33,25,34,12,24,14,22,36,35,13,19,23,20)(10,16,31,39,28,37,17,32,15,26,38,40,18,29,27,30)Polycyclic presentation:
2^10|b^2=f,c^2=f,e^2=fhi,h^2=i,i^2=j,c^a=cg,c^b=cg,d^a=dj,d^b=dg,e^b=ei,e^c=eh,f^e=fj,h^b=hj,h^c=hi,h^e=hj,i^c=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72314270, #72314441, #72314521, … |
| maximal normal subgroup of | #72314270, #72314441, #72314521, … |
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.6);;
gap> SetPower(coll, 5, F.6*F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);