Finite Group DatabasebetaFinite Group Databasebeta| order | 8192 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #72216190 |
| commutator | C162 | C22×C8 |
| Frattini, cosocle | #72216182 | C23 |
| prime radical | C23 | #72214006 |
| LCS | G ▷ C162 ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×C82 ◁ #72216182 ◁ G | |
| LECS | G ▷ #72216182 ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
1413 conjugacy classes of subgroups; only the 530 normal subgroups are shown.
This group has 1413 conjugacy classes of subgroups — too many to draw. Only its 530 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) = 1024 |
A minimal faithful permutation representation — 3 generators on 64 points:
(2,7,8,17,9,18,21,20)(3,11,12,25)(4,14)(5,16)(10,24,22,19,23,31,30,32)(13,28,27,26)(15,29)(34,39,40,49,41,50,53,52)(35,43,44,57)(36,46)(37,48)(42,56,54,51,55,63,62,64)(45,60,59,58)(47,61);(1,2,3,7,4,10,11,19,5,8,13,20,14,22,26,24,6,9,12,18,15,23,25,32,16,21,27,17,29,30,28,31);(33,34,35,39,36,42,43,51,37,40,45,52,46,54,58,56,38,41,44,50,47,55,57,64,48,53,59,49,61,62,60,63)Polycyclic presentation:
2^13|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,h^2=j,i^2=k,j^2=l,k^2=m,b^a=be,c^a=cf,d^b=djl,d^c=dkm,e^a=eh,e^d=el,f^a=fi,f^d=fm,g^b=gl,g^c=gm,h^a=hj,i^a=ik,j^a=jl,k^a=km344 conjugacy classes — view the full character table.
| maximal subgroup of | #72225329 |
| maximal normal subgroup of | #72225329 |
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", "k", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.13);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.13);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);