Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152186 |
| commutator | #492 | C22×C4 |
| Frattini, cosocle | #3147 | C23 |
| prime radical | #18480 | C22 |
| LCS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G | |
| LECS | G ▷ #3147 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
976 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 976 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,13,18,47,19,48,30,59)(4,20)(5,25)(6,8)(7,29)(9,16,40,53,33,54,43,58)(10,24)(11,27)(12,28)(14,46,17,55,50,63,56,62)(15,37)(21,44)(22,60)(23,42)(26,45)(31,34)(32,35)(36,51,61,64,38,52,39,49)(41,57);(2,8)(3,14,18,17,19,50,30,56)(4,21)(9,36,40,61,33,38,43,39)(11,35)(12,29)(13,46,47,55,48,63,59,62)(15,23)(16,51,53,64,54,52,58,49)(22,57)(24,31)(25,45);(1,3,15,51,6,19,23,52)(2,9,37,63,8,33,42,46)(4,16,26,56,21,54,5,17)(7,30,22,49,28,18,57,64)(10,13,35,61,34,48,11,39)(12,43,41,55,29,40,60,62)(14,24,58,32,50,31,53,27)(20,59,45,36,44,47,25,38)Polycyclic presentation:
2^10|a^2=gi,b^2=gi,c^2=dfj,e^2=h,f^2=j,g^2=ij,h^2=j,b^a=bi,c^a=cd,c^b=ce,d^a=dh,d^b=dg,e^a=eghj,e^c=eh,e^d=eh,f^a=fh,f^b=fgj,f^e=fhj,g^c=gh,g^d=gij,g^f=gij,h^a=hi,h^c=hj,h^d=hj,h^f=hj,i^c=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72283540, #72284107 |
| maximal normal subgroup of | #72283540, #72284107 |
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*F.9);;
gap> SetPower(coll, 2, F.7*F.9);;
gap> SetPower(coll, 3, F.4*F.6*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, 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, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);