Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #12090 |
| commutator, Frattini, cosocle | #1438 | C23 |
| prime radical | #244499 | C2 |
| LCS | G ▷ #1438 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1438 ◁ G | |
| LECS | G ▷ #1438 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
1078 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,8,15)(3,10)(4,12)(9,16,14,11)(18,22,24,27)(23,30,31,32)(25,28);(1,2,5,8)(3,7,10,15)(4,11,12,16)(6,9,13,14)(17,18,21,24)(19,23,26,31)(20,27,29,22)(25,30,28,32);(1,2)(3,11)(4,7)(5,8)(6,14)(9,13)(10,16)(12,15)(17,19,21,26)(18,23,24,31)(20,28,29,25)(22,30,27,32)Polycyclic presentation:
2^10|a^2=h,b^2=i,c^2=j,d^2=fi,g^2=ij,b^a=bd,c^a=ce,c^b=cij,d^a=df,d^b=dfi,d^c=dfgij,e^a=eij,e^b=efgh,e^d=ehi,f^c=fj,g^a=gh,g^b=gj,g^c=gi,g^d=gi,g^e=gij,h^b=hi,h^c=hij49 conjugacy classes — view the full character table.
| maximal quotient of | #72267813 |
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.8);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 7, F.9*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.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);