Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #137261 |
| commutator | C2×M32 | C22×C4 |
| Frattini, cosocle | C4×M32 | C23 |
| prime radical | #13309 | C4 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #33418 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(1,2)(3,13,14,41,16,40,28,43)(4,18,20,53)(5,23)(6,27)(7,9)(8,15,30,48,31,47,38,44)(10,21,33,57)(11,25)(12,29)(17,51,45,42,50,35,24,62)(19,37,56,32,55,64,49,60)(22,39,58,54)(26,61)(34,52)(36,46,63,59);(1,3)(2,8)(4,19)(5,24)(6,28)(7,16)(9,31)(10,35)(11,37)(12,38)(13,39)(14,29)(15,46)(17,52)(18,41)(20,55)(21,48)(22,56)(23,60)(25,50)(26,45)(27,30)(32,61)(33,51)(34,64)(36,62)(40,54)(42,63)(43,53)(44,57)(47,59)(49,58);(2,9)(10,33)(11,34)(12,27)(13,40)(15,47)(18,53)(19,55)(23,61)(35,51)(36,63)(39,54)(41,43)(42,62)(44,48)(49,56)Polycyclic presentation:
2^10|a^2=d,d^2=g,f^2=h,h^2=i,i^2=j,b^a=bf,c^a=cj,c^b=ce,d^b=dij,e^a=egj,f^a=fh,f^b=fhij,f^c=fg,f^d=fj,f^e=fj,g^b=gj,h^a=hi,h^b=hij,i^a=ij,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72343113 |
| maximal normal subgroup of | #72343113 |
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, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*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, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);