Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#479 ⋊ C2 |
| Sylow | C5 ⋊ #3119 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#479 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #479 | D5 |
| hypocenter | C5 | #3119 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #479 | |
396 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|b^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^4,d^b=dg,h^b=h^4,f^c=fg,h^c=h^4,h^d=h^470 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3119, C2×#10612939, #12967675, … |
| maximal normal subgroup of | D5×#3119, C2×#10612939, #12967675, … |
| maximal quotient of | #12099417, #12102386, #12102414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15021);