Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #18203 |
| commutator, Frattini, cosocle | C2×#97 | C24 |
| prime radical | #7613571 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,16,20,30)(12,15)(13,26)(17,31,19,28)(18,29,34,40)(21,24)(22,35)(25,27)(32,39,33,38)(36,37);(2,4)(3,6)(7,8)(9,10,13,16,15,19,26,31,14,20,27,30,12,17,25,28)(11,18,22,29,21,33,35,39,23,34,36,40,24,32,37,38);(1,3,5,8)(2,6,4,7)(10,17,20,19)(13,27)(16,28,30,31)(18,33,34,32)(22,36)(25,26)(29,39,40,38)(35,37);(3,8)(6,7)(9,11)(10,18)(12,24)(13,22)(14,23)(15,21)(16,29)(17,32)(19,33)(20,34)(25,37)(26,35)(27,36)(28,38)(30,40)(31,39)Polycyclic presentation:
2^10|a^2=gh,b^2=fhj,c^2=gh,e^2=h,f^2=i,i^2=j,b^a=bf,c^a=cgh,c^b=ce,d^c=dgh,e^a=eg,e^b=egh,e^c=ehj,e^d=ej,f^a=fi,f^c=fg,f^e=fj,g^b=gj,i^a=ij67 conjugacy classes — view the full character table.
| maximal subgroup of | #72307699, #72307707, #72307766 |
| maximal normal subgroup of | #72307699, #72307707, #72307766 |
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*F.8);;
gap> SetPower(coll, 2, F.6*F.8*F.10);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);