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 | |
3003 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,24)(20,27)(21,28)(22,36)(23,37)(25,33)(26,34)(29,30)(31,35)(32,39)(38,40);(1,3,6,2)(4,7,5,8)(9,11,13,10)(12,22)(14,29,27,35)(15,18,28,21)(16,25)(17,31,20,30)(19,36)(23,34)(24,33)(26,32)(37,40)(38,39);(1,4,6,5)(2,7,3,8)(10,11)(12,23)(14,30)(15,28)(16,32)(17,35)(19,37)(20,29)(22,38)(24,39)(25,40)(26,36)(27,31)(33,34);(1,4)(2,7)(3,8)(5,6)(10,11)(16,19)(17,20)(18,21)(22,33)(29,35)(32,37)(34,38);(1,5,6,4)(2,7,3,8)(9,12,14,25,15,26,31,39,13,24,27,36,28,40,30,23)(10,16,17,33,18,34,35,32,11,19,20,22,21,38,29,37)Polycyclic presentation:
2^10|b^2=f,c^2=gj,e^2=ghj,h^2=i,i^2=j,b^a=bg,c^a=cj,c^b=cg,d^a=dj,d^b=dg,d^c=dgj,e^a=egj,e^b=eh,e^c=ehij,e^d=egj,f^e=fj,h^b=hi,h^c=hij,i^b=ij,i^c=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72314475, #72314521, #72314574, … |
| maximal normal subgroup of | #72314475, #72314521, #72314574, … |
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.10);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
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.10);;
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, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);