Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #7734 |
| commutator, Frattini, cosocle | C4×#95 | C23 |
| prime radical | #449474 | C2 |
| LCS | G ▷ C4×#95 ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C4×#95 ▷ C23×C4 ▷ C22 ▷ C1 | |
2029 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 2029 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10,14,18)(11,15,21,30)(12,16,23,32)(13,25,27,17)(19,35,37,40)(20,34,26,29)(22,33,24,31)(28,38,39,36);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,26)(14,21)(16,28)(17,34)(18,30)(20,27)(22,38)(23,37)(24,36)(25,29)(31,40)(32,39)(33,35);(3,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,28)(17,33)(18,32)(20,38)(21,37)(22,27)(25,31)(26,36)(29,40)(30,39)(34,35)Polycyclic presentation:
2^10|a^2=i,d^2=f,e^2=fg,h^2=j,b^a=bd,c^a=ce,d^a=df,d^b=df,d^c=dfhj,e^a=efg,e^b=efhi,e^c=efg,e^d=eij,f^c=fj,g^b=gj,g^c=gj,h^a=hi,h^b=hj58 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267791 |
| maximal subgroup of | #72267791 |
| maximal normal subgroup of | #72267791 |
| maximal quotient of | #72267804, #72267848 |
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.9);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);