Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D16 |
| commutator | C16 | C24×C4 |
| Frattini, cosocle | C2×C16 | C25 |
| prime radical | C2×#33982 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ C4×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
2575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,4,3)(7,17)(8,22)(9,23)(10,26)(13,32)(14,36)(15,37)(16,31)(18,25)(19,39)(20,47)(21,48)(24,44)(27,51)(28,52)(29,33)(30,55)(34,62)(35,63)(38,61)(40,41)(42,46)(43,65)(45,66)(49,50)(53,68)(54,59)(56,64)(57,58)(60,67);(5,6)(7,13)(8,14)(9,15)(10,16)(11,12)(17,32)(18,33)(19,28)(20,34)(21,35)(22,36)(23,37)(24,38)(25,29)(26,31)(27,30)(39,52)(40,57)(41,58)(42,59)(43,60)(44,61)(45,53)(46,54)(47,62)(48,63)(49,56)(50,64)(51,55)(65,67)(66,68);(6,12)(13,28)(14,29)(15,30)(16,31)(32,52)(33,36)(34,53)(35,54)(37,55)(38,56)(57,67)(58,60)(59,63)(61,64)(62,68);(5,7,18,43,9,20,24,42,26,46,50,47,51,65,8,17,11,19,22,41,27,45,49,48,10,21,44,66,23,40,25,39)(6,13,33,60,15,34,38,59,31,54,64,62,55,67,14,32,12,28,36,58,30,53,56,63,16,35,61,68,37,57,29,52);(1,3,4,2)(7,19)(13,28)(17,39)(20,45)(21,46)(32,52)(34,53)(35,54)(40,65)(41,43)(42,48)(47,66)(57,67)(58,60)(59,63)(62,68)Polycyclic presentation:
2^10|a^2=g,d^2=fhj,e^2=g,f^2=hi,h^2=ij,i^2=j,c^b=cj,d^a=df,e^d=ej,f^a=fh,h^a=hi,i^a=ij184 conjugacy classes — view the full character table.
| maximal subgroup of | #72232013, #72321422, #72321439, … |
| maximal normal subgroup of | #72232013, #72321422, #72321439, … |
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.7);;
gap> SetPower(coll, 4, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, 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.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);