Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #97550 |
| commutator | C26 | C22×C4 |
| Frattini, cosocle | C24×D4 | C23 |
| prime radical | #6499870 | C2 |
| LCS | G ▷ C26 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C24 ▷ C2 ▷ C1 | |
3211 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 3211 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,14)(12,19)(13,20)(15,21)(16,24)(17,23)(18,25)(22,27)(26,29)(28,31)(30,32);(1,3,5,13)(2,7,9,20)(4,15,16,30)(6,14,18,29)(8,21,22,32)(10,11,25,26)(12,27,28,17)(19,24,31,23);(1,4)(2,8)(3,12,14,15)(5,16)(6,17)(7,19,11,21)(9,24)(10,23)(13,28,29,30)(18,27)(20,32,26,31)(22,25)Polycyclic presentation:
2^10|b^2=d,c^2=h,b^a=be,c^a=ch,c^b=cf,d^a=dg,d^c=di,e^b=eg,e^c=eh,e^d=ej,f^a=fhj,f^b=fi,f^c=fj,g^b=gj,g^c=gj,h^b=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72237578, #72237580, #72237581, … |
| maximal normal subgroup of | #72237578, #72237580, #72237581, … |
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.4);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
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, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);