Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C5 ⋊ #2762 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, prime radical, cosocle | C23×C5 | C24 |
| Fitting | C5×#383 | C2 |
| hypercenter | #383 | D5 |
| hypocenter | C5 | #2762 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #383 | |
352 conjugacy classes of subgroups.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=f,b^2=e,c^2=fg,d^2=e,b^a=be,c^a=cfg,d^a=defg,h^a=h^4,c^b=cg,d^b=de,h^c=h^4,h^d=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #12307286, #12307292, #12307536, … |
| maximal normal subgroup of | #12307286, #12307292, #12307536, … |
| maximal quotient of | #12026882, #12026905, #12027130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 8096);