Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13674 |
| commutator, Frattini, cosocle | #493 | C24 |
| prime radical | #512858 | C2 |
| LCS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1411 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 1411 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 32 points:
(1,2,8,3)(4,15,18,31,19,30,25,28)(5,14,22,10)(6,11,7,12)(9,26,27,21,13,17,16,32)(20,24,29,23);(1,3,8,2)(4,16)(5,20,22,29)(6,11,7,12)(9,18)(10,23,14,24)(13,25)(15,32)(17,28)(19,27)(21,30)(26,31);(2,3)(9,13)(10,14)(11,12)(15,30)(16,27)(20,29)(28,31);(1,4)(2,9)(3,13)(5,21)(6,18)(7,25)(8,19)(10,28)(11,27)(12,16)(14,31)(15,29)(17,23)(20,30)(22,32)(24,26)Polycyclic presentation:
2^10|a^2=ghi,b^2=i,e^2=h,g^2=i,h^2=j,c^a=cj,c^b=cj,d^a=de,d^b=df,e^a=ej,e^b=eg,e^d=ehj,f^a=fgh,f^b=fj,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72306759, #72306902, #72338366, … |
| maximal normal subgroup of | #72306759, #72306902, #72338366, … |
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*F.9);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);