Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | Q8×D52⋊C2 |
| Fitting | C52×Q8 ⋊ D4 |
| Sylow | C52 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D52⋊C2 |
| commutator, cosocle | C2×C5⋊D5 | C24 |
| Fitting | C52×Q8 | D4 |
| socle | C2×C52 | C22×D4 |
| prime radical | C2×D52⋊C2 | C22 |
| hypercenter | Q8 | D52⋊C2 |
| hypocenter | C52 | D4×Q8 |
| LCS | G ▷ C2×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
471 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 10037);