Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #8350 |
| commutator | C2×M16 | C23×C4 |
| Frattini, cosocle | C2×#131 | C24 |
| socle | C2 | C2×#8350 |
| prime radical | #10577631 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
3992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,9)(3,15,20,19)(4,21)(5,26)(10,34,31,36)(13,14)(16,41,37,52)(17,42,23,48)(18,49,51,43)(25,30)(28,29)(35,56,59,60)(38,44)(45,50,62,63)(47,64,54,61)(53,57);(1,2)(3,16,20,37)(4,22)(5,27)(6,14)(7,9)(8,13)(10,17,31,23)(11,25)(12,28)(15,41,19,52)(18,50,51,63)(21,40)(24,38)(26,39)(29,33)(30,32)(34,42,36,48)(35,61,59,64)(43,45,49,62)(44,46)(47,60,54,56)(53,58)(55,57);(2,9)(10,31)(11,32)(12,33)(13,14)(16,37)(22,40)(27,39)(34,36)(35,59)(38,44)(41,52)(45,62)(50,63)(53,57)(56,60);(1,3,5,18,8,19,29,49,7,20,26,51,6,15,28,43)(2,10,12,35,14,34,39,60,9,31,33,59,13,36,27,56)(4,23,24,54,25,42,55,61,21,17,46,47,30,48,58,64)(11,37,38,62,22,52,53,50,32,16,44,45,40,41,57,63)Polycyclic presentation:
2^10|a^2=hj,b^2=hj,d^2=f,e^2=i,f^2=i,g^2=j,i^2=j,b^a=bh,c^b=cj,d^a=dg,d^b=de,e^a=eh,e^b=ei,e^d=ej,f^a=fj,f^b=fij,g^b=gh,g^e=gj,h^d=hj,i^b=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72307207, #72307431, #72307535 |
| maximal normal subgroup of | #72307207, #72307431, #72307535 |
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.8*F.10);;
gap> SetPower(coll, 2, F.8*F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
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.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, 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);