Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C5×#469 ⋊ C2 |
| Sylow | C5 ⋊ #3058 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#469 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #469 | D5 |
| hypocenter | C5 | #3058 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #469 | |
524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|b^2=eg,e^2=g,b^a=be,c^a=cf,e^a=eg,h^b=h^4,d^c=dg,h^d=h^464 conjugacy classes — view the full character table.
| maximal subgroup of | #12010292, #12010300, #12253200, … |
| maximal normal subgroup of | #12010292, #12010300, #12253200, … |
| maximal quotient of | #12082369, #12082408, #12082456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 14678);