Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C52 · Schur multiplier (H2) = C23×C5 |
| direct product | C52×#450 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C52 |
| prime radical | C22×C52×D4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ C22×C4×C52 ◁ G | |
752 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 400 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5)(11,12)(13,14)(15,16,18,21)(17,19,22,20);(6,7,8,9,10)(16,19)(17,22)(20,21);(12,14)475 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5539);