Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C5 ⋊ C4×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator | C22×C5 | C23×C4 |
| Fitting | C5×#389 | C2 |
| socle, prime radical, cosocle | C23×C5 | C24 |
| hypercenter | #389 | D5 |
| hypocenter | C5 | C4×C4⋊Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #389 | |
588 conjugacy classes of subgroups.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=g,b^2=e,c^2=f,d^2=eg,b^a=be,c^a=cf,h^b=h^4,h^d=h^4112 conjugacy classes — view the full character table.
| maximal subgroup of | #11987276, #11987358, #11987371, … |
| maximal normal subgroup of | #11987276, #11987358, #11987371, … |
| maximal quotient of | #12020490, #12020492, Dic5×#389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 9780);