Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
797 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 797 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 64 points:
(1,16,24,59,8,43,20,63)(2,31,46,60,13,52,35,58)(3,23,50,26,18,19,40,54)(4,39,7,61,5,49,29,42)(6,17,47,22,27,45,51,15)(9,48,38,36,33,34,41,53)(10,44,12,64,21,37,25,56)(11,32,57,14,28,55,62,30);(1,2,4,10,8,13,5,21)(3,14,15,9,18,30,22,33)(6,25,19,46,27,12,23,35)(7,28,20,48,29,11,24,34)(16,41,39,55,43,38,49,32)(17,44,40,52,45,37,50,31)(26,53,47,62,54,36,51,57)(42,60,59,64,61,58,63,56)Polycyclic presentation:
2^4,3,2^5|a^2=c,b^2=cj,c^2=j,d^2=j,b^a=bd,d^a=dj,d^b=dj,e^a=e^2,e^b=e^2,f^a=ghj,f^b=g,f^e=g,g^a=fhij,g^b=f,g^e=fg,h^a=hi,h^b=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij56 conjugacy classes — view the full character table.
| maximal subgroup of | #72342100 |
| maximal normal subgroup of | #72342100 |
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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3*F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);