Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C52 · Schur multiplier (H2) = C22×C5 |
| direct product | C52×#436 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C52 | C23 |
| commutator | C22 | C22×C4×C52 |
| Frattini, cosocle | C2×C4 | C23×C52 |
| socle | C22×C52 | C22×C4 |
| prime radical | C2×C52×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C52 ◁ G | |
488 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 296 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 — 3 generators on 22 points:
(1,2,3,4,5)(11,12)(13,14)(15,16,19,17,18,20,22,21);(6,7,8,9,10)(12,14)(16,20)(17,21);(15,17,19,20,18,21,22,16)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5525);