Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | Q82×D5 |
| Fitting | C5×Q82 ⋊ C2 |
| Sylow | C5 ⋊ C2×Q82 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×Q82 | C2 |
| prime radical | C22×D5 | C24 |
| hypercenter | Q82 | D5 |
| hypocenter | C5 | C2×Q82 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 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 | D5×#1623, D5×#1633, D5×#1645, … |
| maximal normal subgroup of | D5×#1623, D5×#1633, D5×#1645, … |
| maximal quotient of | Q8×D5×C4⋊C4, D5×#2369, Q8×#4033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21028);