Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #486654 |
| commutator, Frattini, cosocle | C2×M32 | C24 |
| prime radical | #7624863 | C2 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #33434 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,12,25,44,53,29,7,10,26,35,6,14,24,9,8,13,4,19,22,37,55,33,27,36,23,32,28,45,20,31)(3,11,15,41,48,61,64,56,17,34,49,59,16,43,46,30,18,42,21,40,47,60,63,57,50,39,54,58,51,62,52,38);(2,9)(4,20)(5,24)(6,22)(7,27)(10,29)(11,38)(12,14)(13,31)(15,46)(16,47)(17,50)(19,45)(21,52)(23,55)(25,28)(26,53)(30,42)(32,37)(33,36)(34,57)(35,44)(39,56)(40,43)(41,62)(48,51)(49,64)(54,63)(58,61)(59,60);(2,10,13,36)(9,29,31,33)(11,39,42,34)(12,32,19,35)(14,37,45,44)(30,57,38,56)(40,58,41,59)(43,61,62,60);(1,3)(2,11)(4,21)(5,15)(6,16)(7,17)(8,18)(9,30)(10,34)(12,41)(13,42)(14,43)(19,40)(20,52)(22,47)(23,54)(24,46)(25,48)(26,49)(27,50)(28,51)(29,56)(31,38)(32,58)(33,57)(35,59)(36,39)(37,60)(44,61)(45,62)(53,64)(55,63)Polycyclic presentation:
2^10|a^2=f,c^2=g,e^2=hi,f^2=hij,h^2=ij,i^2=j,b^a=be,c^a=cefghij,d^b=dg,d^c=dg,e^a=eg,e^b=eh,e^c=eg,e^d=ej,f^b=fgh,f^c=fg,f^e=fj,g^a=gj,h^b=hi,i^b=ij73 conjugacy classes — view the full character table.
| maximal subgroup of | #72320889, #72321336, #72342980 |
| maximal normal subgroup of | #72320889, #72321336, #72342980 |
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.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);