Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #151342 |
| commutator, Frattini, cosocle | #1781 | C23 |
| prime radical | #151130 | C2 |
| LCS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
743 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 743 conjugacy classes of subgroups — too many to draw. Only its 37 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:
(2,7)(3,11)(5,19)(8,23)(9,15)(12,29)(13,28)(14,16)(22,24)(25,32)(26,27)(30,31);(1,2)(3,12)(4,15,18,9)(5,20)(6,7)(8,13,22,31)(10,19)(11,27)(14,26)(16,29)(17,25,21,32)(23,30,24,28);(1,3,6,14)(2,8)(4,16)(5,13)(7,22)(9,23,15,24)(10,26,20,12)(11,18)(17,28,21,30)(19,31)(25,27)(29,32)Polycyclic presentation:
2^10|b^2=fh,c^2=hj,d^2=g,e^2=gh,b^a=bj,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dgi,e^a=efhj,e^b=eg,e^c=eghij,e^d=ehj,f^a=fi,f^b=fij,f^c=fh,f^d=fi,f^e=fij,g^b=gi,g^e=gj,h^a=hi,h^b=hij,h^d=hj,h^e=hj,i^c=ij34 conjugacy classes — view the full character table.
| maximal subgroup of | #72283898, #72283924, #72284072, … |
| maximal normal subgroup of | #72283898, #72283924, #72284072, … |
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*F.8);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);