Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #2297 |
| commutator | C22×C4 | C22×C42 |
| Frattini, cosocle | C23×C8 | C24 |
| socle | C2 | #354290 |
| prime radical | C2×#525 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,7,12,26,39,8,13,6,27,25,40,28,55)(3,14,16,42,19,9,29,31,17,33,18,47,46,32,48,38)(4,10,21,35,24,37,52,60,22,36,23,53,51,61,54,64)(15,41,43,62,45,30,56,57,20,49,44,63,50,34,58,59);(1,3,8,17)(2,9)(4,20,22,15)(5,18,6,16)(7,19,25,46)(10,34)(11,38)(12,33)(13,32)(14,40)(21,43,23,44)(24,50,51,45)(26,48,28,29)(27,31)(30,36)(35,57)(37,41)(39,42)(47,55)(49,61)(52,56,54,58)(53,59)(60,63)(62,64);(1,4)(2,10)(3,15)(5,21)(6,23)(7,24)(8,22)(9,30)(11,35)(12,37)(13,36)(14,41)(16,43)(17,20)(18,44)(19,45)(25,51)(26,52)(27,53)(28,54)(29,56)(31,57)(32,34)(33,49)(38,59)(39,60)(40,61)(42,62)(46,50)(47,63)(48,58)(55,64);(15,20)(30,34)(41,49)(43,44)(45,50)(56,58)(57,59)(62,63)Polycyclic presentation:
2^10|a^2=e,b^2=ehij,e^2=i,h^2=i,i^2=j,b^a=bij,c^b=cj,d^b=df,d^c=dg,e^b=ej,f^c=fj,g^b=gj,h^a=hj,h^b=hj160 conjugacy classes — view the full character table.
| maximal subgroup of | #72343936, #72343946, #72343971, … |
| maximal normal subgroup of | #72343936, #72343946, #72343971, … |
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.5);;
gap> SetPower(coll, 2, F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);