Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C5×#389 ⋊ C2 |
| Sylow | C5 ⋊ #2808 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#389 | C2 |
| prime radical | C22×D20 | C22 |
| hypercenter | #389 | D5 |
| hypocenter | C5 | #2808 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #389 | |
428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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,b^2=f,c^2=fg,d^2=efg,b^a=beg,c^a=cf,d^a=defg,c^b=cg,d^b=deg,h^b=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #11988062, #11988066, #11988230, … |
| maximal normal subgroup of | #11988062, #11988066, #11988230, … |
| maximal quotient of | #12027763, #12027772, #12028063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 9998);