Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#1812 |
| commutator | C22×C8 | C23×C4 |
| Frattini, cosocle | C2×#131 | C24 |
| socle | C2 | #188809 |
| prime radical | #3444 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C4×C8 ◁ #33421 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
1117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8)(3,13)(9,25)(10,26)(12,28)(14,37)(16,39)(17,40)(18,41)(22,50)(30,33)(32,51)(42,44)(45,61)(46,55)(56,58);(1,2,4,11)(3,14,16,43)(5,22,20,48)(6,25,21,49)(7,8,19,27)(9,29,31,23)(10,32,34,17)(12,36,35,15)(13,37,39,60)(18,42,46,62)(24,50,47,64)(26,51,52,40)(28,54,53,38)(30,56,57,61)(33,58,59,45)(41,44,55,63);(2,9,8,25)(3,13)(10,33,26,30)(11,31,27,49)(12,22,28,50)(14,42,37,44)(16,39)(17,40)(18,41)(32,51)(34,59,52,57)(35,48,53,64)(43,62,60,63)(45,61)(46,55)(56,58);(1,3,15,45,6,18,24,40,7,13,38,61,23,41,5,17)(2,10,22,44,9,30,28,37,8,26,50,42,25,33,12,14)(4,16,36,58,21,46,47,51,19,39,54,56,29,55,20,32)(11,34,48,63,31,57,53,60,27,52,64,62,49,59,35,43)Polycyclic presentation:
2^10|b^2=e,c^2=h,d^2=gi,g^2=ij,i^2=j,b^a=bf,c^b=cij,d^a=dhj,d^b=dg,d^c=dj,e^a=eh,f^b=fh,f^d=fj,f^e=fj,g^b=gi,h^b=hj,i^b=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72287373, #72287420, #72287462, … |
| maximal normal subgroup of | #72287373, #72287420, #72287462, … |
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.5);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);