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 | #7622822 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ #20384 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
4151 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)(10,15)(16,19)(17,33)(18,31)(23,38)(24,37)(30,34)(36,39);(2,5)(3,8)(9,10,13,15)(11,16,21,19)(12,23)(14,30,29,34)(17,32)(18,26,31,22)(20,39)(24,25)(27,33)(28,38)(35,36)(37,40);(1,3,4,8)(2,6,5,7)(9,10)(11,19)(12,24,28,37)(13,15)(14,31)(16,21)(17,35,33,20)(18,29)(22,34)(23,40,38,25)(26,30)(27,36,32,39);(2,5)(6,7)(9,11)(10,16)(12,25,28,40)(13,21)(14,29)(15,19)(17,36,33,39)(20,32,35,27)(23,24,38,37)(30,34);(2,5)(6,7)(9,12,26,40,21,35,29,27,13,28,22,25,11,20,14,32)(10,17,34,36,16,24,31,23,15,33,30,39,19,37,18,38)Polycyclic presentation:
2^10|b^2=f,c^2=fh,d^2=fj,e^2=fgi,g^2=i,i^2=j,b^a=bh,c^a=ch,c^b=ch,d^a=dhj,d^c=dj,e^a=ehj,e^b=eg,e^c=eij,e^d=egj,f^e=fj,g^b=gi,g^c=gj,g^d=gi,g^e=gj,i^b=ij,i^d=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72314371, #72314441, #72315390, … |
| maximal normal subgroup of | #72314371, #72314441, #72315390, … |
| maximal quotient of | #72343772 |
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.6*F.8);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*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, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);