Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#66050 |
| Fitting | C2×C5×#127 ⋊ C2 |
| Sylow | C5 ⋊ C2×#532 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D5 |
| commutator, cosocle | C22×C5 | C25 |
| Frattini | C22 | C24×D5 |
| Fitting | C2×C5×#127 | C2 |
| socle | C23×C5 | C24 |
| hypercenter | C2×#127 | D5 |
| hypocenter | C5 | C2×#532 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
1728 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 557 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 | #11990789, #11990802, #12014790, … |
| maximal normal subgroup of | #11990789, #11990802, #12014790, … |
| maximal quotient of | C2×#10606049, C2×#10608020, C2×#10608021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 20170);