Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139700 |
| commutator | C2×#127 | C22×C4 |
| Frattini, cosocle | C2×#535 | C23 |
| prime radical | C2×#3021 | C22 |
| LCS | G ▷ C2×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1716 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,8)(4,18)(5,10)(6,25)(7,12)(9,22)(11,26)(13,38)(14,28)(15,30)(16,37)(17,29)(19,27)(20,45)(21,32)(23,36)(24,34)(31,49)(33,46)(35,51)(39,61)(40,58)(41,60)(42,55)(43,53)(44,54)(47,52)(48,56)(50,59)(57,64)(62,63);(1,3)(2,8)(4,19,46,40)(5,14)(6,16,26,15)(7,17)(9,27,32,58)(10,28)(11,37,25,30)(12,29)(13,21,43,22)(18,38,33,53)(20,48,64,39)(23,42)(24,41,51,44)(31,56,59,61)(34,54,35,60)(36,55)(45,63,57,52)(47,49,62,50);(1,4,5,20,7,21,23,50)(2,9,10,31,12,33,36,57)(3,13,14,39,17,40,42,62)(6,22,24,49,26,46,51,64)(8,27,28,52,29,53,55,56)(11,18,35,45,25,32,34,59)(15,43,44,48,16,19,41,47)(30,38,54,61,37,58,60,63)Polycyclic presentation:
2^10|b^2=fg,c^2=d,d^2=j,h^2=j,b^a=bf,c^a=cghj,c^b=ce,d^b=dg,e^a=eghj,e^b=egh,e^c=eg,e^d=ej,f^c=fgh,f^e=fij,g^c=gj,h^a=hj,h^b=hi,h^e=hj,h^f=hj,i^a=ij,i^b=ij,i^c=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72318343, #72318472, #72319038, … |
| maximal normal subgroup of | #72318343, #72318472, #72319038, … |
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.6*F.7);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);