Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#442 |
| commutator | C2×C8 | C24×C4 |
| Frattini, cosocle | C4×C8 | C25 |
| prime radical | #7623522 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
2254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,15)(14,24)(16,23)(17,21)(18,25)(19,26)(20,27)(22,28)(29,32)(30,31)(35,43)(39,52)(44,59)(45,61)(46,51)(49,63)(53,55)(54,64);(1,3)(2,8)(4,17)(5,14)(6,16)(7,15)(9,10)(11,22)(12,13)(18,30)(19,28)(20,29)(21,24)(23,26)(25,27)(31,32)(33,34,37,42,38,40,50,57,36,41,47,58,48,56,62,60)(35,44,45,39,46,54,49,55,43,59,61,52,51,64,63,53);(1,4,6,18,7,14,19,29)(2,9,11,25,12,24,26,32)(3,5,16,20,15,17,28,30)(8,21,22,31,13,10,23,27)(33,35,36,43)(34,39)(37,49,47,63)(38,51,48,46)(40,53)(41,52)(42,59)(44,58)(45,62,61,50)(54,57)(55,56)(60,64);(1,5,19,30,7,17,6,20)(2,10,26,31,12,21,11,27)(3,14,28,18,15,4,16,29)(8,9,23,32,13,24,22,25)(34,40,41,56)(37,47)(39,53,52,55)(42,60,58,57)(44,54,59,64)(45,61)(49,63)(50,62);(3,15)(4,14)(8,13)(9,24)(16,28)(18,29)(22,23)(25,32)(34,41)(39,52)(40,56)(42,58)(44,59)(53,55)(54,64)(57,60)Polycyclic presentation:
2^10|b^2=g,c^2=f,d^2=fhj,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=cj,c^b=cghij,d^b=dij,e^b=ehj,e^c=eh,f^b=fj,g^c=gi,g^d=gj,i^c=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72319103, #72319381 |
| maximal normal subgroup of | #72319103, #72319381 |
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.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6*F.8*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);