Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C5×#382 ⋊ C2 |
| Sylow | C5 ⋊ #2352 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator | C22×C5 | C23×C4 |
| Fitting | C5×#382 | C2 |
| socle, cosocle | C23×C5 | C24 |
| prime radical | C22×D20 | C22 |
| hypercenter | #382 | D5 |
| hypocenter | C5 | #2352 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #382 | |
660 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=g,b^2=f,c^2=f,b^a=be,c^a=cf,d^b=de,h^b=h^4,h^d=h^4106 conjugacy classes — view the full character table.
| maximal subgroup of | #12114714, #12114716, #12114740, … |
| maximal normal subgroup of | #12114714, #12114716, #12114740, … |
| maximal quotient of | #12020716, #12020724, #12020748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 8042);