Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D5×#396 |
| Fitting | C5×#396 ⋊ C2 |
| Sylow | C5 ⋊ C2×#396 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#396 | C2 |
| prime radical | C24×D5 | C22 |
| hypercenter | #396 | D5 |
| hypocenter | C5 | C2×#396 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #396 | |
908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D5×#396, D5×#2392, D5×#2396, … |
| maximal normal subgroup of | C2×D5×#396, D5×#2392, D5×#2396, … |
| maximal quotient of | D5×#1439, D5×#1441, D5×#1443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 11226);