Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
5988 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 5988 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:
(1,24,5,2,15,10)(3,32,13,11,28,25)(4,9)(6,26,17,8,18,22)(7,31,20,14,23,30)(12,27)(16,21)(19,29);(2,7,8,11)(4,15,16,18)(9,23)(10,20,22,25)(12,28,29,31)(19,26)(21,32)(24,27)Polycyclic presentation:
2^4,3,2^5|a^2=c,d^2=j,h^2=j,i^2=j,b^a=bd,c^b=cj,d^b=dj,e^a=e^2,f^a=fg,f^e=g,g^e=fg,h^a=hi,h^e=i,h^g=hj,i^a=ij,i^e=hij,i^f=ij,i^h=ij62 conjugacy classes — view the full character table.
| maximal subgroup of | #72227648, #72227841, #72252442, … |
| maximal normal subgroup of | #72227648, #72227841, #72252442, … |
| maximal quotient of | #72268131, #72268245, #72268982, … |
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, 4, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);