Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #129029 |
| commutator | C2×C16 | C2×C42 |
| Frattini, cosocle | C4×M32 | C23 |
| prime radical | #2218 | C2×C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #11406 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(2,8,11,25)(3,14,17,45,18,22,53,47)(4,20,16,41)(5,24)(6,27)(9,26)(10,33,35,44,37,50,38,61)(12,40)(13,43,39,30)(15,36,49,59,28,42,51,64)(19,55,54,46,48,63,31,57)(21,34)(23,32,52,58)(29,62);(1,2,9,30,27,39,26,8,7,11,24,43,6,13,5,25)(3,15,48,35,17,51,31,10,18,28,19,38,53,49,54,37)(4,21,58,62,52,34,20,56,16,40,32,29,23,12,41,60)(14,44,46,36,47,50,55,59,22,61,57,42,45,33,63,64);(1,3,16,14,26,48,20,57,6,17,52,47,9,31,58,63,7,18,4,22,5,19,41,46,27,53,23,45,24,54,32,55)(2,10,34,36,25,15,29,50,13,38,21,59,43,51,56,61,11,37,12,42,8,28,62,33,39,35,40,64,30,49,60,44)Polycyclic presentation:
2^10|a^2=d,b^2=dghij,c^2=ef,d^2=g,e^2=h,h^2=i,i^2=j,b^a=bgh,c^a=ce,c^b=cgh,d^b=dj,d^c=dij,e^a=eh,e^b=eij,e^c=eg,e^d=ej,f^a=fg,f^c=fg,f^e=fj,g^c=gj,h^a=hi,h^b=hj,i^a=ij70 conjugacy classes — view the full character table.
| maximal subgroup of | #72319087, #72343434, #72343458 |
| maximal normal subgroup of | #72319087, #72343434, #72343458 |
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.4);;
gap> SetPower(coll, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);