Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #6346637 |
| commutator, Frattini, cosocle | C2×M16 | C25 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,17)(4,11)(5,12)(6,13)(7,30)(8,15)(9,16)(10,20)(14,32)(18,46)(19,35)(21,36)(22,45)(23,54)(24,39)(25,40)(26,41)(27,42)(28,44)(29,33)(31,37)(34,49)(38,55)(43,50)(47,51)(48,59)(52,63)(53,62)(56,64)(57,60)(58,61);(1,3,8,20)(2,10,15,17)(4,6,9,29)(5,23)(7,28,32,50)(11,33,16,13)(12,38)(14,44,30,43)(18,48)(19,22,51,31)(21,52)(24,56)(25,49)(26,58)(27,55)(34,59)(35,37,47,45)(36,61)(39,62)(40,46)(41,63)(42,54)(53,57)(60,64);(1,4,8,9)(2,11,15,16)(3,6,20,29)(7,31,32,22)(10,33,17,13)(14,45,30,37)(18,49)(19,50,51,28)(24,57)(25,48)(34,46)(35,43,47,44)(39,60)(40,59)(53,56)(62,64);(1,5,7,25,9,26,31,56,8,27,32,48,4,21,22,53)(2,12,14,40,16,41,45,64,15,42,30,59,11,36,37,62)(3,18,19,23,6,24,28,52,20,49,51,55,29,57,50,58)(10,34,35,38,33,60,44,61,17,46,47,54,13,39,43,63);(1,6,9,20,8,29,4,3)(2,13,16,10,15,33,11,17)(5,24,26,49,27,57,21,18)(7,28,31,51,32,50,22,19)(12,39,41,34,42,60,36,46)(14,43,45,35,30,44,37,47)(23,25,52,56,55,48,58,53)(38,59,61,62,54,40,63,64)Polycyclic presentation:
2^10|b^2=h,c^2=h,d^2=f,e^2=i,f^2=i,i^2=j,b^a=bj,d^a=dg,d^b=df,d^c=di,e^b=ei,e^c=ej,f^a=fh,f^b=fi,f^c=fj,g^b=ghj,g^d=gh,g^f=gj,h^d=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72250237, #72250830, #72281944, … |
| maximal normal subgroup of | #72250237, #72250830, #72281944, … |
| maximal quotient of | #72250381, #72280113 |
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.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);