Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C5 |
| direct product | #125×C5⋊D5 |
| Fitting | C52×#125 ⋊ C2 |
| Sylow | C52 ⋊ C2×#125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C52×#125 | C2 |
| prime radical | C22×C5⋊D5 | C23 |
| hypercenter | #125 | C5⋊D5 |
| hypocenter | C52 | C2×#125 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ #125 | |
904 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 904 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 800 |
196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9568);