Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152049 |
| commutator | C4×M16 | C22×C4 |
| Frattini, cosocle | #3151 | C23 |
| prime radical | #6358039 | C2 |
| LCS | G ▷ C4×M16 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C4×#41 ◁ #60817 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1210 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 81 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)(3,14)(4,10)(5,12)(6,11)(7,13)(8,17)(9,22)(15,25)(16,23)(18,26)(19,29)(20,27)(21,28)(24,31)(30,32);(1,3,4,16)(2,8,10,25)(5,18,20,31)(6,17,21,15)(7,19,22,30)(9,24,13,26)(11,29,28,32)(12,14,27,23);(3,15,18,30,19,31,17,16)(8,23,14,25,26,32,29,24)Polycyclic presentation:
2^10|b^2=e,c^2=gh,d^2=hi,f^2=j,g^2=ij,h^2=j,b^a=bf,c^b=cd,d^a=dij,d^b=dg,e^c=egh,e^d=ehi,f^a=fj,f^b=fj,f^c=fi,f^d=fj,g^b=gh,g^e=gij,h^b=hi,h^e=hj,i^b=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72339838, #72339887, #72339889 |
| maximal normal subgroup of | #72339838, #72339887, #72339889 |
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.5);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);