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 | C22 | #10336 |
| commutator, Frattini, cosocle | C22×#127 | C23 |
| prime radical | #245233 | C2 |
| LCS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
2361 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 2361 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,15)(4,12)(5,8)(6,10,14,16)(7,13)(17,18)(19,25,27,22)(20,23,29,30)(21,24)(26,31)(28,32);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,21,27)(18,22,24,25)(20,28,29,31)(23,26,30,32);(7,12)(10,16)(17,20,21,29)(18,23,24,30)(19,26,27,32)(22,28,25,31)Polycyclic presentation:
2^10|a^2=gh,b^2=j,c^2=j,f^2=j,b^a=bd,c^a=ce,c^b=cefj,d^a=dij,d^c=dg,e^a=ej,e^b=eh,e^d=ei,f^a=fghj,f^b=fh,f^d=fj,f^e=fj,g^a=gi,g^b=gij,h^a=hi,h^c=hj46 conjugacy classes — view the full character table.
| maximal subgroup of | #72337638, #72337658, #72337660, … |
| maximal normal subgroup of | #72337638, #72337658, #72337660, … |
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.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);