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 | #10576932 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
3641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(3,13)(9,29)(14,39)(15,42)(17,25)(32,34)(44,46)(48,52);(1,2)(3,14,13,39)(4,18)(5,23)(6,28)(7,8)(9,17,29,25)(10,35)(11,38)(12,20)(15,46,42,44)(16,49,43,63)(19,37)(21,40)(22,30)(24,41)(26,36)(27,31)(32,48,34,52)(33,60,59,64)(45,50,62,47)(51,61,53,54)(55,58)(56,57);(2,8)(3,15,13,42)(4,19)(5,24)(9,32,29,34)(12,28)(14,44,39,46)(16,50,43,47)(17,52,25,48)(22,35)(27,38)(33,61,59,54)(36,40)(45,63,62,49)(51,64,53,60)(55,57);(1,3,4,16,20,42,35,50,7,13,19,43,6,15,22,47)(2,9,10,33,28,32,18,54,8,29,30,59,12,34,37,61)(5,25,26,51,38,48,56,60,24,17,21,53,27,52,58,64)(11,39,40,62,23,44,57,49,31,14,36,45,41,46,55,63)Polycyclic presentation:
2^10|b^2=h,c^2=h,d^2=e,e^2=ij,f^2=ij,i^2=j,c^b=cj,d^a=dg,d^b=df,d^c=di,e^a=eh,e^b=ehij,e^c=ej,f^b=fij,f^c=fj,f^d=fh,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, #72307980, #72308079 |
| maximal normal subgroup of | #72307146, #72307980, #72308079 |
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*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*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, 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*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
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);