Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139906 |
| commutator | C23×Q8 | C22×C4 |
| Frattini, cosocle | #2310 | C23 |
| prime radical | C22×#41 | C22⋊C4 |
| LCS | G ▷ C23×Q8 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35316 ◁ G | |
| LECS | G ▷ #2310 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 869 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:
(1,2,4,9)(3,13,15,43,17,44,47,58)(5,21,18,50)(6,25,19,34)(7,12,20,35)(8,28,30,49,31,56,32,16)(10,36,33,24)(11,39,40,27)(14,42,45,63,22,54,51,64)(23,38,52,61)(26,41,53,60)(29,55,57,46,37,62,59,48);(3,14,17,22)(8,29,31,37)(13,42,44,54)(15,45,47,51)(16,48,49,46)(28,55,56,62)(30,57,32,59)(43,63,58,64);(1,3)(2,8,12,31)(4,15)(5,22)(6,16)(7,17)(9,32,35,30)(10,37,21,29)(11,13,25,44)(14,24)(18,51)(19,28)(20,47)(23,48)(26,46)(27,49)(33,57,50,59)(34,43,40,58)(36,45)(38,42,41,54)(39,56)(52,55)(53,62)(60,63,61,64)Polycyclic presentation:
2^10|a^2=d,b^2=h,c^2=i,d^2=h,e^2=j,f^2=j,c^a=cf,c^b=ce,d^c=dg,e^a=ehj,e^c=ej,f^a=fgj,f^b=fh,f^d=fi,f^e=fj,g^a=gi,g^d=gj,h^c=hj,i^a=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72284297 |
| maximal normal subgroup of | #72284297 |
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, 2, F.8);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
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);