Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C5 |
| direct product | M32×C5⋊D5 |
| Fitting | C52×M32 ⋊ C2 |
| Sylow | C52 ⋊ C2×M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×C5⋊D5 |
| commutator, socle | C2×C52 | C22×C8 |
| Fitting | C52×M32 | C2 |
| prime radical | C22×C5⋊D5 | C8 |
| cosocle | C52×C8 | C23 |
| hypercenter | M32 | C5⋊D5 |
| hypocenter | C52 | C2×M32 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C8 ◁ M32 | |
360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
280 conjugacy classes. The character table is too large to display.
| maximal subgroup of | M32×#1043, C5⋊D5×#347, #72242137, … |
| maximal normal subgroup of | M32×#1043, C5⋊D5×#347, #72242137, … |
| maximal quotient of | C5⋊D5×#346, C2×M32×C5⋊D5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5121);