Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #18186 |
| commutator, Frattini, cosocle | C2×#97 | C24 |
| prime radical | #7613946 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,20,25,37)(12,17)(13,28)(14,26)(15,19)(16,24,33,40)(18,31)(21,35,32,39)(22,36,23,38)(27,29)(30,34);(1,2,3,8,5,7,4,6)(9,11,13,24,14,23,31,39,15,25,30,40,12,22,27,35)(10,16,18,20,17,21,28,36,19,33,29,37,26,32,34,38);(2,7)(6,8)(9,10)(11,21,25,32)(12,26)(13,29)(14,17)(15,19)(16,23,33,22)(18,30)(20,35,37,39)(24,38,40,36)(27,28)(31,34);(2,6)(3,4)(7,8)(11,22,25,23)(13,30)(16,21,33,32)(18,29)(20,36,37,38)(24,35,40,39)(27,31)(28,34)Polycyclic presentation:
2^10|a^2=gh,b^2=fh,c^2=gh,d^2=gh,e^2=hj,f^2=i,i^2=j,b^a=bf,c^b=cij,d^b=de,d^c=dgh,e^a=eg,e^b=egh,e^c=ej,e^d=eh,f^a=fi,f^c=fj,f^d=fgj,f^e=fj,g^b=gj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72307615, #72307699, #72307751 |
| maximal normal subgroup of | #72307615, #72307699, #72307751 |
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);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.8*F.10);;
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, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*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.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.10);;
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);