Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #110180 |
| commutator | C25 | C2×C42 |
| Frattini, prime radical, cosocle | C22×H(5,𝔽2) | C23 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
1599 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1599 conjugacy classes of subgroups — too many to draw. Only its 151 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)(3,13,15,42)(4,18,20,47)(6,11,24,38)(7,12,25,39)(8,29,31,26)(9,33,35,22)(14,41,44,62)(16,32,46,37)(17,43,23,53)(19,49,50,64)(21,36,52,60)(27,48,54,63)(28,40,55,61)(30,56,57,51)(34,58,59,45);(1,3)(2,8,12,32)(4,19,22,14)(5,23,25,15)(6,26,28,16)(7,17)(9,34,18,30)(10,37)(11,13)(20,50)(21,51)(24,46)(27,45)(29,55)(31,39)(33,44)(35,59)(36,49)(38,53,61,42)(40,43)(41,48)(47,57)(52,56,54,58)(60,64,63,62);(1,4)(2,9,12,18)(3,14)(5,20)(6,27,28,21)(7,22)(8,30)(10,35,39,47)(11,36)(13,41,43,49)(15,44)(16,45,26,51)(17,19)(23,50)(24,54,55,52)(25,33)(29,56,46,58)(31,57)(32,34)(37,59)(38,60)(40,48)(42,62,53,64)(61,63)Polycyclic presentation:
2^10|a^2=d,b^2=f,c^2=gh,b^a=be,c^a=ch,c^b=cfg,d^b=dh,e^a=eh,e^b=ehi,e^c=ehi,f^a=fhi,f^c=fij,f^d=fj,f^e=fj,g^a=gj,g^b=gij,g^c=gj,g^e=gj,h^b=hj,h^c=hj,i^a=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72284743, #72284859, #72284938, … |
| maximal normal subgroup of | #72284743, #72284859, #72284938, … |
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.6);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*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.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);