Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13086 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C2×#127 | C24 |
| prime radical | #6437453 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ C2×#3500 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
4062 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)(3,14)(4,9)(5,10)(6,11)(7,13)(8,18)(12,26)(15,43)(16,42)(17,30)(19,32)(20,34)(21,40)(22,36)(23,38)(24,37)(25,29)(27,53)(28,50)(31,46)(33,52)(35,49)(39,55)(41,45)(44,62)(47,61)(48,58)(51,60)(54,59)(56,63)(57,64);(1,3)(2,8,11,30,13,29,12,14)(4,15,20,45)(5,16)(6,25)(7,17)(9,31,33,43,34,27,40,41)(10,35,37,42,38,28,39,58)(18,26)(19,44,22,47)(21,53,52,46)(23,48)(24,49)(32,56,59,61,36,57,60,62)(50,55)(51,63,54,64);(1,4)(2,9)(3,15)(5,22)(6,21)(7,20)(8,27)(10,36)(11,40)(12,33)(13,34)(14,41)(16,47)(17,45)(18,46)(19,23)(24,54)(25,53)(26,52)(28,57)(29,31)(30,43)(32,38)(35,56)(37,59)(39,60)(42,62)(44,48)(49,64)(50,63)(51,55)(58,61);(1,5)(2,10)(3,16)(4,19)(6,24)(7,23)(8,28)(9,32)(11,37)(12,39)(13,38)(14,42)(15,44)(17,48)(18,50)(20,22)(21,51)(25,49)(26,55)(27,56)(29,35)(30,58)(31,57)(33,59)(34,36)(40,60)(41,61)(43,62)(45,47)(46,64)(52,54)(53,63)Polycyclic presentation:
2^10|b^2=f,f^2=ij,h^2=j,b^a=bg,c^a=ce,c^b=cei,d^b=dij,d^c=dj,e^b=ej,f^a=fh,f^c=fj,g^b=ghj,g^d=gj,g^f=gij,h^a=hj,h^b=hi,h^g=hj,i^a=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72317859, #72318143, #72319309 |
| maximal normal subgroup of | #72317859, #72318143, #72319309 |
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.6);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);