Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#2196 |
| commutator, Frattini, cosocle | #391 | C24 |
| prime radical | C2×#23890 | C2 |
| LCS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
3640 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 3640 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(11,20)(12,15)(13,26)(16,30)(17,25)(18,36)(21,22)(27,39)(28,29)(31,37)(32,40)(35,38);(9,10)(11,16)(12,25)(13,18)(14,19)(15,17)(20,30)(21,40)(22,32)(23,33)(24,34)(26,36)(27,37)(28,38)(29,35)(31,39);(9,11,14,20)(10,16,19,30)(12,23,15,24)(13,27,28,22)(17,34,25,33)(18,37,38,32)(21,29,39,26)(31,36,40,35);(1,2,4,6)(3,8,7,5)(9,12,14,15)(10,17,19,25)(11,21)(13,26,28,29)(16,31)(18,35,38,36)(20,39)(22,23)(24,27)(30,40)(32,34)(33,37)Polycyclic presentation:
2^10|c^2=j,d^2=hj,e^2=j,f^2=h,g^2=j,c^a=cj,d^a=df,d^b=dj,d^c=de,e^a=eg,e^c=ej,f^a=fh,f^c=fgi,f^d=fh,f^e=fi,g^a=gj,g^d=gi,g^e=gj,h^c=hj,i^c=ij67 conjugacy classes — view the full character table.
| maximal subgroup of | #72340688 |
| maximal normal subgroup of | #72340688 |
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, 3, F.10);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, 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, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);