Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C52×#123 ⋊ C22 |
| Sylow | C52 ⋊ C2×#458 |
1930 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1930 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 26 points:
(2,3)(4,5)(6,7)(8,9);(1,2,4,5,3)(11,14,15,21)(12,17,18,25)(13,19,20,26)(16,22,23,24);(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(7,8)(9,10)(11,12,15,18)(13,16,20,23)(14,17,21,25)(19,24,26,22)Polycyclic presentation:
2^7,5^2|a^2=g,c^2=g,c^a=cf,c^b=ce,e^a=eg,f^b=fg,h^a=h^4,h^d=h^4,i^a=h^3i,i^d=i^4152 conjugacy classes — view the full character table.
| maximal subgroup of | #72226507 |
| maximal normal subgroup of | #72226507 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);