Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #24128 |
| commutator, Frattini, cosocle | C22×C8 | C25 |
| socle | C2 | C2×#24128 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
5014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,9)(4,17,19,46)(5,22)(6,26)(7,8)(10,20,29,51)(11,24)(12,27)(13,28)(14,39,37,64)(15,43)(16,45)(18,47,42,62)(21,54,40,48)(23,56)(25,55)(30,58)(31,41,60,57)(32,44)(33,38)(34,52,61,50)(35,63,59,53)(36,49);(2,8)(9,28)(10,29)(11,30)(12,26)(17,46)(22,55)(31,60)(32,36)(33,45)(34,61)(35,59)(39,64)(43,56)(47,62)(48,54);(1,3)(2,9)(4,18,19,42)(5,23)(6,16)(7,13)(8,28)(10,34,29,61)(11,36)(12,33)(14,40,37,21)(15,25)(17,47,46,62)(20,52,51,50)(22,56)(24,49)(26,45)(27,38)(30,32)(31,59,60,35)(39,48,64,54)(41,53,57,63)(43,55)(44,58);(3,13)(9,28)(15,23)(16,38)(32,36)(33,45)(43,56)(44,49);(1,4,5,20,6,21,24,53,7,19,25,51,27,40,58,63)(2,10,11,17,12,35,22,48,8,29,30,46,26,59,55,54)(3,14,15,41,16,42,44,52,13,37,23,57,38,18,49,50)(9,31,32,39,33,61,43,47,28,60,36,64,45,34,56,62)Polycyclic presentation:
2^10|a^2=g,c^2=g,e^2=h,h^2=i,i^2=j,b^a=bj,d^c=dgj,e^a=eh,e^c=eij,e^d=ef,f^c=fj,g^e=gj,h^a=hi,h^c=hj,i^a=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72307129 |
| maximal normal subgroup of | #72307129 |
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, 3, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);