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 | #7623961 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ #20521 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
2702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(10,15)(11,19,22,38)(12,23)(13,26)(16,30,20,32)(17,34)(21,37,27,24)(25,39)(28,36)(31,40,33,35);(1,2,4,5)(3,7,8,6)(9,10)(11,20)(12,17)(13,18)(14,15)(16,22)(19,30)(21,31)(23,36)(24,40)(25,28)(26,29)(27,33)(32,38)(34,39)(35,37);(3,8)(6,7)(9,11,13,27,14,22,26,21)(10,16,18,31,15,20,29,33)(12,24,23,19,25,37,39,38)(17,35,36,32,28,40,34,30);(2,5)(3,8)(10,15)(11,21,22,27)(12,25)(16,31,20,33)(18,29)(19,37,38,24)(23,39)(30,40,32,35);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=fi,d^2=f,e^2=g,h^2=i,i^2=j,b^a=bj,c^a=ch,c^b=cj,d^b=dgj,d^c=dij,e^b=eg,e^c=eg,e^d=eg,f^c=fj,h^a=hi,h^c=hi,h^d=hj,i^a=ij,i^c=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72313898, #72314098, #72314251, … |
| maximal normal subgroup of | #72313898, #72314098, #72314251, … |
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.6);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);