Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #146566 |
| commutator | C22×Q8 | C2×C42 |
| Frattini, cosocle | C22×#124 | C23 |
| prime radical | C2×#3500 | C22 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
2876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,8)(4,17,19,50)(5,23,24,38)(6,26,27,12)(7,10)(9,21,34,54)(11,25)(13,39,40,61)(14,32)(15,45,43,60)(16,47,44,31)(18,49,42,62)(20,51,22,52)(28,37)(29,41,57,63)(30,46)(33,53,58,64)(35,55,36,56)(48,59);(1,3)(2,8)(4,18)(5,15)(6,16)(7,14)(9,33)(10,32)(11,30)(12,31)(13,20)(17,49)(19,42)(21,53)(22,40)(23,45)(24,43)(25,46)(26,47)(27,44)(28,48)(29,35)(34,58)(36,57)(37,59)(38,60)(39,51)(41,55)(50,62)(52,61)(54,64)(56,63);(1,4)(2,9,10,35)(3,13,14,18)(5,21)(6,19)(7,20)(8,29)(11,17,37,51)(12,34,26,36)(15,41,43,53)(16,40,44,42)(22,27)(23,50,38,52)(24,55)(25,56)(28,54)(30,39)(31,57)(32,33)(45,61)(46,64,48,63)(47,58)(49,59)(60,62)Polycyclic presentation:
2^10|a^2=d,c^2=e,g^2=j,h^2=j,c^a=cg,c^b=cf,d^c=dh,e^a=ej,e^b=ej,f^c=fj,g^a=gh,g^d=gij,h^a=hi,h^c=hj,h^d=hj,h^g=hj,i^a=ij,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72318053, #72318077, #72318119, … |
| maximal normal subgroup of | #72318053, #72318077, #72318119, … |
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.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);