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 | #7623621 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ D4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3242 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)(3,7)(4,5)(6,8);(1,3,5,8)(2,6,4,7)(9,10,14,18)(11,19,23,15)(12,16,21,30)(13,17,28,36)(20,37,38,29)(22,33,25,31)(24,35,27,32)(26,34,39,40);(2,4)(3,8)(11,20,23,38)(12,26)(13,28)(15,29,19,37)(16,34)(17,36)(21,39)(22,24,25,27)(30,40)(31,32,33,35);(1,4,5,2)(3,7,8,6)(9,11,21,24,13,25,39,38,14,23,12,27,28,22,26,20)(10,15,30,32,17,33,40,37,18,19,16,35,36,31,34,29);(2,4)(6,7)(11,22,23,25)(12,21)(15,31,19,33)(16,30)(20,24,38,27)(26,39)(29,32,37,35)(34,40)Polycyclic presentation:
2^10|b^2=gj,c^2=f,d^2=fghj,e^2=f,h^2=i,i^2=j,b^a=bg,c^a=cg,c^b=cg,d^a=dg,d^b=dj,d^c=dghij,e^a=eg,e^d=egi,f^d=fj,h^c=hi,h^d=hj,h^e=hj,i^c=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72314098, #72314160, #72314371, … |
| maximal normal subgroup of | #72314098, #72314160, #72314371, … |
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.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
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.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
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);