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 | |
3114 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 3114 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) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(10,13)(11,15)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26,29,32)(27,31,28,30);(1,2,4,6)(3,7,8,5)(11,15)(13,16)(18,22)(19,20)(23,24)(25,27,29,28)(26,30)(31,32);(1,3,4,8)(2,5,6,7)(9,10,12,14)(11,16,15,13)(17,19,21,20)(18,23,24,22)(26,30,32,31)Polycyclic presentation:
2^10|a^2=j,b^2=f,c^2=fgh,d^2=g,e^2=h,b^a=be,c^a=cd,c^b=cfhj,d^a=dg,d^b=dgi,d^c=dgj,e^a=eh,e^b=ehj,e^c=egi,e^d=ej,f^a=fj,i^a=ij88 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267774 |
| maximal subgroup of | #72267774 |
| maximal normal subgroup of | #72267774 |
| maximal quotient of | #72267779 |
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.10);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.7);;
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.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);