Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
504 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 32 points:
(9,19,32)(10,27,29,15,24,21)(11,13,22)(12,23,26)(14,25,28,17,16,31)(18,20,30);(1,2,3,5,4,6,7,8)(9,10,11,14,12,15,18,17)(13,21,19,28,20,29,23,31)(16,26,24,30,25,32,27,22)Polycyclic presentation:
2^4,3,2^5|a^2=c,c^2=d,f^2=i,g^2=h,b^a=bhij,e^a=e^2,e^b=ehi,f^a=gh,f^c=fij,f^e=gij,g^a=fj,g^c=ghj,g^e=fgj,g^f=gj,h^a=i,h^e=hij,i^a=h,i^e=h60 conjugacy classes — view the full character table.
| maximal subgroup of | #72267671 |
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, 3, F.4);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);