Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
6199 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 6199 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,31,4,14)(2,32,22,28)(3,16,13,5)(6,17,15,30)(7,26,27,10)(8,25,9,21)(11,24,20,23)(12,18,29,19);(1,2)(3,11)(4,10,18,9,5,26)(6,8)(7,12)(13,28,17,27,14,32)(15,23,16,22,19,24)(20,30,25,29,21,31)Polycyclic presentation:
2^4,3,2^5|b^2=d,h^2=j,i^2=j,b^a=bc,c^b=cj,d^a=dj,e^b=e^2,f^b=g,f^e=g,g^b=f,g^e=fg,h^b=fgh,h^e=i,h^f=hj,h^g=hj,i^b=ghij,i^e=hij,i^g=ij,i^h=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72227958, #72227989, #72241394, … |
| maximal normal subgroup of | #72227958, #72227989, #72252367, … |
| maximal quotient of | C2×#72227957, #72268131, #72268140 |
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, 2, F.4);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);