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 | #7623625 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ D4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,3)(4,7,5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,30)(20,37)(21,32)(22,28)(23,33)(24,34)(25,26)(27,29)(31,38)(35,39)(36,40);(1,3,6,2)(4,7,5,8)(9,10)(11,19,22,34)(12,25)(13,18)(14,15)(16,30,28,24)(17,26)(20,33,38,36)(21,35)(23,31,40,37)(27,29)(32,39);(1,4)(2,7)(3,8)(5,6)(10,15)(11,20,22,38)(12,21)(13,27)(16,31,28,37)(17,35)(19,36,34,33)(23,24,40,30)(25,32)(26,39);(2,3)(4,5)(10,15)(16,28)(17,25)(18,29)(19,34)(31,37)(32,35)(33,36);(1,5,6,4)(2,8,3,7)(9,11,21,38,27,30,12,23,14,22,39,20,13,24,26,40)(10,16,32,31,29,19,17,33,15,28,35,37,18,34,25,36)Polycyclic presentation:
2^10|a^2=g,b^2=f,c^2=fg,e^2=ghi,h^2=i,i^2=j,c^a=cj,c^b=cgj,d^a=dgj,d^b=dgj,d^c=dg,e^b=egij,e^c=eghij,e^d=eg,f^e=fj,h^b=hj,h^c=hi,i^c=ij109 conjugacy classes — view the full character table.
| maximal subgroup of | #72314098, #72314160, #72314604, … |
| maximal normal subgroup of | #72314098, #72314160, #72314604, … |
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.7);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 5, F.7*F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
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, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);