Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #10390 |
| commutator, Frattini, cosocle | C22×#127 | C23 |
| prime radical | #35527 | C22 |
| LCS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
1241 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 1241 conjugacy classes of subgroups — too many to draw. Only its 49 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)(3,12)(4,14)(5,9)(6,10)(7,13)(8,15)(11,16)(19,25)(22,30)(26,27)(31,32);(1,3,5,8)(2,7,9,4)(6,13)(10,15)(11,12)(14,16)(17,18)(19,22)(20,28,29,23)(21,24)(25,30,26,31)(27,32);(1,4,5,7)(2,8,9,3)(6,12)(10,13)(11,14)(15,16)(17,19,21,27)(18,22,24,32)(20,25)(23,30)(26,29)(28,31)Polycyclic presentation:
2^10|b^2=d,c^2=dj,e^2=ij,f^2=i,b^a=bf,c^a=cdeij,c^b=cdi,d^a=dg,e^a=eg,e^b=eh,e^c=eg,e^d=eij,f^a=fi,f^b=fgi,f^c=fghij,f^d=fi,g^b=gi,g^c=gij,h^a=hi,h^b=hij,h^c=hj40 conjugacy classes — view the full character table.
| maximal subgroup of | #72338788, #72338801, #72338806, … |
| maximal normal subgroup of | #72338788, #72338801, #72338806, … |
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.4*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.5*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*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, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);