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 | #6357952 | C2 |
| LCS | G ▷ C4×M16 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C22×C42 ◁ #60801 ◁ G | |
| LECS | G ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1194 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)(3,11)(4,9)(5,10)(6,7)(8,13)(12,22)(14,24)(15,23)(16,19)(17,26)(18,20)(21,29)(25,30)(27,31)(28,32);(2,7)(3,12,15,27,13,29,14,28)(4,16)(8,21,24,32,11,22,23,31)(10,20)(17,30);(1,3)(2,8,7,11)(4,12,17,27)(5,15,18,14)(6,13)(9,22,25,32)(10,24)(16,29,30,28)(19,21,26,31)(20,23)Polycyclic presentation:
2^10|b^2=gij,c^2=e,f^2=hj,g^2=ij,h^2=j,b^a=bi,c^a=cd,c^b=cf,d^b=dij,d^c=dj,e^a=ej,e^b=eg,f^a=fij,f^b=fj,f^c=fgh,f^d=fj,f^e=fh,g^c=gh,g^e=gij,h^c=hi,h^e=hj,i^c=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72306790, #72306869, #72306930, … |
| maximal normal subgroup of | #72306790, #72306869, #72306930, … |
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.7*F.9*F.10);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
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);