Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #22858 |
| commutator | C4×C8 | C23×C4 |
| Frattini, cosocle | C2×C4×C8 | C24 |
| prime radical | #33580 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #8203 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,12,15)(11,19)(13,27,25,34)(14,30,26,18)(16,28)(17,21,32,29)(20,37)(22,35)(23,31)(24,36)(33,39)(38,40);(2,5)(3,7)(6,8)(10,15)(11,20,22,24)(13,25)(16,33,31,40)(18,30)(19,36,35,37)(21,29)(23,39,28,38);(1,2,4,6)(3,8,7,5)(10,15)(16,31)(17,32)(18,30)(19,35)(27,34)(33,40)(36,37);(9,11,21,39,26,20,25,23,12,22,29,38,14,24,13,28)(10,16,34,37,30,40,32,19,15,31,27,36,18,33,17,35)Polycyclic presentation:
2^10|a^2=e,b^2=ej,c^2=h,d^2=egi,f^2=h,g^2=i,i^2=j,b^a=bj,c^a=cf,c^b=cfj,d^a=dg,d^b=dij,e^d=ej,f^a=fh,f^b=fh,f^c=fh,g^a=gi,g^b=gj,g^d=gj,i^a=ij85 conjugacy classes — view the full character table.
| maximal subgroup of | #72309705, #72309851, #72309941, … |
| maximal normal subgroup of | #72309705, #72309851, #72309941, … |
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.5);;
gap> SetPower(coll, 2, F.5*F.10);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.5*F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, 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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);