Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×#95 | C23 |
| prime radical | C2×#2656 | C22 |
| LCS | G ▷ C22×#95 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C22×#95 ◁ G | |
| LECS | G ▷ C22×#95 ▷ C24 ▷ C2 ▷ C1 | |
2658 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 2658 conjugacy classes of subgroups — too many to draw. Only its 191 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:
(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,33)(32,37)(34,38)(35,40)(36,39);(1,2,4,6)(3,7,8,5)(10,14)(11,12)(15,16)(17,19,21,20)(18,22,24,23)(25,27,29,35)(26,30,33,40)(28,36,37,34)(31,38,32,39);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,15,16,14)(18,23)(19,20)(22,24)(26,31)(28,37)(30,38)(32,33)(34,36)(39,40)Polycyclic presentation:
2^10|b^2=f,c^2=fhi,d^2=i,e^2=h,b^a=be,c^a=cd,c^b=cfhj,d^a=di,d^b=dg,d^c=di,e^a=eh,e^b=eh,e^c=egj,e^d=ej,g^a=gj88 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267755 |
| maximal subgroup of | #72267755 |
| maximal normal subgroup of | #72267755 |
| maximal quotient of | #72267761 |
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*F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);