Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #2992 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting, prime radical | C5×#121 | C4 |
| cosocle | C2×Dic5 | C24 |
| hypercenter | #121 | AGL(1,𝔽5) |
| hypocenter | C5 | #2992 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #121 | |
356 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=f,d^2=e,f^2=g,b^a=be,d^a=de,h^a=h^2,c^b=ce,d^b=dg,h^d=h^4,h^f=h^458 conjugacy classes — view the full character table.
| maximal subgroup of | #13018956, #13018958, #13018961, … |
| maximal normal subgroup of | #13018956, #13018958, #13018961, … |
| maximal quotient of | #13033956, #13034006, #13034039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19327);