Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #149280 |
| commutator | C2×#124 | C22×C4 |
| Frattini, cosocle | C22×#124 | C23 |
| prime radical | C23×D4 | C2×D4 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,9)(4,20,25,52)(5,27)(6,30)(7,32)(8,14)(10,23,41,58)(11,29)(12,31)(13,22)(15,45,47,64)(16,35)(17,50)(18,37)(19,38)(21,54,55,62)(24,42,59,51)(26,40,57,53)(28,36)(33,44)(34,46,61,56)(39,60,63,48)(43,49);(1,3)(2,9)(4,21,26,56)(5,28,29,17)(6,16)(7,33,22,18)(8,19)(10,39,42,45)(11,36,27,50)(12,43)(13,44,32,37)(14,38)(15,24,48,58)(20,34,40,54)(23,47,59,60)(25,55,57,46)(30,35)(31,49)(41,63,51,64)(52,61,53,62);(1,4,22,57,8,25,7,26)(2,10,13,42,14,41,32,51)(3,15,18,48,19,47,33,60)(5,23,31,59,29,58,6,24)(9,34,44,62,38,61,37,54)(11,20,30,53,27,52,12,40)(16,46,28,55,49,56,17,21)(35,45,50,63,43,64,36,39)Polycyclic presentation:
2^10|a^2=f,b^2=g,c^2=dei,d^2=ij,e^2=i,h^2=j,b^a=bg,c^a=cd,c^b=ce,d^a=di,d^b=dh,e^a=ej,e^b=ehi,f^c=fj,g^c=ghj,g^d=gi,g^e=gij,h^a=hi,h^b=hij,h^g=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72286360, #72286456, #72286670 |
| maximal normal subgroup of | #72286360, #72286456, #72286670 |
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, 2, F.7);;
gap> SetPower(coll, 3, F.4*F.5*F.9);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);