Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#4090 |
| Sylow | C5 ⋊ C22×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D5 |
| commutator, cosocle | C22×C5 | C25 |
| Frattini | C22 | C24×D5 |
| Fitting | C23×C5×Q8 | C2 |
| socle, prime radical | C24×C5 | C23 |
| hypercenter | C23×Q8 | D5 |
| hypocenter | C5 | C22×C4⋊Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C24 ◁ C23×Q8 | |
1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 640 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | #11991932, #12005850, #12005851, … |
| maximal normal subgroup of | #11991932, #12005850, #12005851, … |
| maximal quotient of | C22×#65265, C22×#65261, C22×#65263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21229);