Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #150393 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #1781 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2,5,10)(3,13,15,41)(4,18,19,34)(6,11,22,38)(7,12,23,36)(8,26,28,16)(9,30,32,21)(14,42,43,64)(17,37,46,62)(20,47,49,61)(24,29,53,56)(25,39,51,35)(27,54,55,52)(31,58,59,44)(33,48,60,50)(40,57,63,45);(1,3,7,17)(2,8)(4,14,21,45)(5,15,23,46)(6,24,25,16)(9,31)(10,28)(11,37)(12,29)(13,39)(18,27)(19,43,30,57)(20,44,50,52)(22,26,51,53)(32,55)(33,40)(34,59)(35,41)(36,56)(38,62)(42,47)(48,58,49,54)(60,64)(61,63);(1,4,7,21)(2,9,12,18)(3,14,17,45)(5,19)(6,20)(8,27)(10,34)(11,33)(13,40,37,42)(15,43)(16,44)(22,48,51,49)(23,30)(24,52)(25,50)(26,54,53,58)(28,55,56,59)(29,31)(32,36)(35,61,38,60)(39,47)(41,64)(46,57)(62,63)Polycyclic presentation:
2^10|a^2=d,b^2=ij,c^2=ghj,e^2=g,b^a=be,c^a=cfgj,c^b=cf,d^b=df,d^c=dhij,e^a=efg,e^b=egj,e^c=eh,e^d=ehij,f^a=fh,f^c=fi,f^e=fij,g^a=gij,g^c=gj,g^d=gj,h^b=hi,h^c=hj,h^e=hj,i^a=ij43 conjugacy classes — view the full character table.
| maximal subgroup of | #72283898, #72284175, #72285226, … |
| maximal normal subgroup of | #72283898, #72284175, #72285226, … |
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.9*F.10);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);