Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | Q8×#713 |
| Fitting | C5×Q82 ⋊ C2 |
| Sylow | C5 ⋊ Q8×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×Q82 | C2 |
| prime radical | C2×#713 | C22 |
| hypercenter | Q82 | D5 |
| hypocenter | C5 | Q8×#41 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1068 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 529 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | #12075275, #12075283, #12075290, … |
| maximal normal subgroup of | #12075275, #12075283, #12075290, … |
| maximal quotient of | C4⋊C4×#713, Q8×#4038, #12258956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21022);