Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | Dic5×#127 |
| Sylow | C5 ⋊ C4×#127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator | C22×C5 | C23×C4 |
| Fitting, prime radical | C2×C5×#127 | C2 |
| socle, cosocle | C23×C5 | C24 |
| hypercenter | C2×#127 | D5 |
| hypocenter | C5 | C4×#127 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
924 conjugacy classes of subgroups; only the 289 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 289 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
112 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×#331, #11990255, #11990268, … |
| maximal normal subgroup of | Dic5×#331, #11990255, #11990268, … |
| maximal quotient of | Dic5×#390, Dic5×#384, Dic5×#392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 16984);