Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #93182 |
| commutator | C22×C22⋊C4 | C42 |
| Frattini, cosocle | C2×#2614 | C22 |
| prime radical | C23×D4 | C22⋊C4 |
| LCS | G ▷ C22×C22⋊C4 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2614 ◁ G | |
| LECS | G ▷ C2×#2614 ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 41 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 — 2 generators on 64 points:
(1,2,4,9)(3,13,14,37,17,39,41,55)(5,21,22,49)(6,11,19,32)(7,12,20,33)(8,26,27,43,29,50,30,15)(10,18,31,24)(16,44,45,61,46,38,40,62)(23,35,51,60)(25,36,48,54)(28,53,57,42,58,63,56,64)(34,47,59,52);(1,3,6,16,7,17,25,46)(2,8,11,28,12,29,36,58)(4,14,19,40,20,41,48,45)(5,15,23,42,24,43,52,64)(9,30,32,57,33,27,54,56)(10,13,34,38,21,39,35,44)(18,26,47,53,22,50,51,63)(31,37,59,61,49,55,60,62)Polycyclic presentation:
2^10|a^2=c,b^2=e,c^2=h,d^2=hi,e^2=j,b^a=bd,c^b=cf,d^a=dfhi,d^b=dg,d^c=dij,e^a=eghi,e^c=eh,e^d=ehi,f^a=fi,f^b=fh,f^c=fj,f^d=fj,f^e=fj,g^a=ghj,g^b=ghi,h^b=hj,i^a=ij,i^b=ij34 conjugacy classes — view the full character table.
| maximal subgroup of | #72268804 |
| maximal normal subgroup of | #72268804 |
| maximal quotient of | #72268870 |
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.3);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*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);