Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Sylow | C52 ⋊ C2×#379 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D52 |
| commutator | C22×C52 | C23×C4 |
| Fitting | C25×C52 | C22 |
| socle, cosocle | C23×C52 | C24 |
| prime radical | C25×C5⋊D5 | C2 |
| hypercenter | C25 | D52 |
| hypocenter | C52 | C2×#379 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C23 ◁ C25 | |
12578 conjugacy classes of subgroups; only the 484 normal subgroups are shown.
This group has 12578 conjugacy classes of subgroups — too many to draw. Only its 484 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 22 points:
(14,15)(16,17)(18,19)(20,21);(6,8)(10,12)(13,14,16,17,15);(6,8);(1,2,3,4)(5,6)(7,8)(9,10,11,12)(19,20)(21,22)Polycyclic presentation:
2^7,5^2|a^2=f,b^a=be,c^a=cefg,h^a=h^4,h^d=h^4,i^a=h^3i,i^d=i^4296 conjugacy classes — view the full character table.
| maximal quotient of | #72226543 |
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.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*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);