Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
1363 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,3,4,7)(5,8)(9,16,38,29)(10,20,40,27)(11,28,24,22)(12,34,21,30)(13,18,33,39)(14,25,36,35)(15,23,32,31)(17,19,37,26);(2,3,5)(6,7,8)(10,15)(11,19)(12,13,26)(16,29,34,28,17,40)(18,30)(20,31,38,33,21,36)(22,39,25,35,32,37)(23,27,24)Polycyclic presentation:
2^3,3,2^6|b^2=c,c^2=ij,g^2=ij,h^2=ij,b^a=bc,c^a=cij,d^b=d^2,e^b=ef,e^d=f,f^d=ef,g^b=hij,g^d=h,g^e=gi,g^f=gi,h^b=gij,h^d=ghij,h^f=hi,h^g=hij45 conjugacy classes — view the full character table.
| maximal subgroup of | #72268154 |
| maximal normal subgroup of | #72268154 |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);