Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #185340 |
| commutator | C2×M16 | C23×C4 |
| Frattini, cosocle | C2×#131 | C24 |
| prime radical | #10580980 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
3468 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)(3,17)(4,11)(5,12)(6,28)(7,14)(8,15)(9,16)(10,20)(13,31)(18,46)(19,35)(21,36)(22,44)(23,54)(24,39)(25,40)(26,41)(27,42)(29,43)(30,37)(32,33)(34,49)(38,55)(45,50)(47,51)(48,59)(52,63)(53,62)(56,64)(57,60)(58,61);(1,3,8,20)(2,10,15,17)(4,7,9,32)(5,23)(6,29,31,50)(11,33,16,14)(12,38)(13,43,28,45)(18,48)(19,22,51,30)(21,52)(24,49)(25,56)(26,58)(27,55)(34,59)(35,37,47,44)(36,61)(39,46)(40,62)(41,63)(42,54)(53,57)(60,64);(1,4,8,9)(2,11,15,16)(3,7,20,32)(6,30,31,22)(10,33,17,14)(13,44,28,37)(18,49)(19,50,51,29)(24,48)(25,57)(34,46)(35,45,47,43)(39,59)(40,60)(53,56)(62,64);(1,5,7,25,9,26,20,49,8,27,32,57,4,21,3,18)(2,12,14,40,16,41,10,34,15,42,33,60,11,36,17,46)(6,24,29,52,30,56,51,55,31,48,50,58,22,53,19,23)(13,39,45,63,44,64,35,38,28,59,43,61,37,62,47,54)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=de,d^c=di,e^a=eh,e^b=ei,e^c=ej,f^b=fi,f^c=fj,g^b=ghj,g^e=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 | #72256824 |
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.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
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, 5, 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);