Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#66068 |
| Fitting | C2×C5×C4⋊Q8 ⋊ C2 |
| Sylow | C5 ⋊ C2×#555 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D5 |
| commutator, cosocle | C22×C5 | C25 |
| Frattini | C22 | C24×D5 |
| Fitting | C2×C5×C4⋊Q8 | C2 |
| socle | C23×C5 | C24 |
| hypercenter | C2×C4⋊Q8 | D5 |
| hypocenter | C5 | C2×#555 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
1240 conjugacy classes of subgroups; only the 501 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 501 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
106 conjugacy classes — view the full character table.
| maximal subgroup of | #11991070, #11991421, #11991425, … |
| maximal normal subgroup of | #11991070, #11991421, #11991425, … |
| maximal quotient of | C2×#10607710, C2×#10606055, C2×#10607716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 20183);