Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C52 · Schur multiplier (H2) = C25×C5 |
| direct product | C52×D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C52 |
| prime radical | C2×C52×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ G | |
1120 conjugacy classes of subgroups; only the 728 normal subgroups are shown.
This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 728 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) = 800 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3,4,5)(11,12,13,14);(6,7,8,9,10)(12,14);(15,16,17,18)(19,22,21,20);(15,19,17,21)(16,20,18,22)625 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9759);