Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
| direct product | D5×M64 |
| Fitting | C5×M64 ⋊ C2 |
| Sylow | C5 ⋊ C2×M64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C22×D5 |
| commutator, socle | C2×C5 | C22×C16 |
| Fitting | C5×M64 | C2 |
| prime radical | C22×D5 | C16 |
| cosocle | C5×C16 | C23 |
| hypercenter | M64 | D5 |
| hypocenter | C5 | C2×M64 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C16 ◁ M64 | |
112 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1398, #12019486, D5×#1400, … |
| maximal normal subgroup of | D5×#1398, #12019486, D5×#1400, … |
| maximal quotient of | D5×#1397, Dic5×M64, #12019476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2246);