Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C5 |
| direct product | C2×#11717517 |
| Fitting | C22×C52×D4 ⋊ C2 |
| Sylow | C52 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×C5⋊D5 |
| commutator, cosocle | C2×C52 | C25 |
| Frattini | C2 | C24×C5⋊D5 |
| Fitting | C22×C52×D4 | C2 |
| socle | C22×C52 | C24 |
| hypercenter | C22×D4 | C5⋊D5 |
| hypocenter | C52 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
3592 conjugacy classes of subgroups; only the 867 normal subgroups are shown.
This group has 3592 conjugacy classes of subgroups — too many to draw. Only its 867 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
274 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×H(5,𝔽2)×C5⋊D5, #72221182, #72221185, … |
| maximal normal subgroup of | C2×H(5,𝔽2)×C5⋊D5, #72221182, #72221185, … |
| maximal quotient of | #72221182, #72221185, #72221186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 10233);