Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C5×#123 ⋊ C2 |
| Sylow | C5 ⋊ #2501 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C2×C5×#123 | C2 |
| prime radical | C2×#4085 | C2 |
| hypercenter | C2×#123 | D5 |
| hypocenter | C5 | #2501 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
708 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=g,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^4,h^d=h^494 conjugacy classes — view the full character table.
| maximal subgroup of | #12152678, #12152681, #12154582, … |
| maximal normal subgroup of | #12152678, #12152681, #12154582, … |
| maximal quotient of | #12021283, #12021529, #12022419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 16672);