Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #2514 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, prime radical, cosocle | C23×C5 | C24 |
| Fitting | C5×#391 | C2 |
| hypercenter | #391 | D5 |
| hypocenter | C5 | #2514 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #391 | |
372 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=fg,c^2=eg,d^2=ef,b^a=bef,c^a=cfg,d^a=def,h^a=h^4,c^b=cg,h^b=h^494 conjugacy classes — view the full character table.
| maximal subgroup of | #12172453, #12172457, #12172513, … |
| maximal normal subgroup of | #12172453, #12172457, #12172513, … |
| maximal quotient of | #12021336, #12021679, #12022335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 10337);