Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #2728 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#394 | C2 |
| prime radical | C22×C5×D4 | C22 |
| hypercenter | #394 | D5 |
| hypocenter | C5 | #2728 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #394 | |
512 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,d^2=e,b^a=bef,c^a=cf,d^a=de,c^b=cg,d^b=def,h^d=h^476 conjugacy classes — view the full character table.
| maximal subgroup of | #12011371, #12011375, #12012059, … |
| maximal normal subgroup of | #12011371, #12011375, #12012059, … |
| maximal quotient of | #12027067, #12027424, #12027673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 10856);