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 | #185645 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C2×#131 | C24 |
| prime radical | #10576933 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
3665 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)(3,13)(4,10)(5,11)(6,12)(7,8)(9,17)(14,30)(15,34)(16,32)(18,31)(19,50)(20,27)(21,29)(22,36)(23,35)(24,39)(25,28)(26,38)(33,43)(37,52)(40,49)(41,54)(42,60)(44,56)(45,59)(46,63)(47,58)(48,55)(51,62)(53,61)(57,64);(2,8)(3,14,17,43)(4,19)(5,24)(6,21)(9,30,13,33)(10,35)(11,37)(15,41,18,45)(16,46,49,64)(20,23)(25,52)(27,50)(28,39)(31,54,34,59)(32,57,40,63)(36,38)(42,44,48,47)(51,53)(55,56,60,58);(3,15,17,18)(4,20)(5,25)(9,31,13,34)(10,27)(11,28)(14,41,43,45)(16,47,49,44)(19,23)(24,52)(30,54,33,59)(32,58,40,56)(35,50)(37,39)(42,64,48,46)(55,63,60,57);(1,3,4,16,21,15,19,44,7,17,20,49,6,18,23,47)(2,9,10,32,29,31,50,56,8,13,27,40,12,34,35,58)(5,14,26,42,52,45,53,64,25,43,22,48,24,41,51,46)(11,33,38,60,37,54,61,57,28,30,36,55,39,59,62,63)Polycyclic presentation:
2^10|b^2=h,c^2=h,d^2=e,e^2=ij,f^2=i,i^2=j,b^a=bj,d^a=dg,d^b=df,d^c=dij,e^a=eh,e^b=ehij,e^c=ej,f^b=fi,f^c=fj,f^d=fhj,f^e=fj,g^d=gh,g^e=gj,h^d=hj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72307146, #72307752, #72307980 |
| maximal normal subgroup of | #72307146, #72307752, #72307980 |
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.5);;
gap> SetPower(coll, 5, F.9*F.10);;
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*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
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, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
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);