Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C5×C32 ⋊ C2 |
| Sylow | C5 ⋊ #2258 |
| subgroup | G/H | |
|---|---|---|
| center | C32 | C2×D5 |
| commutator, socle | C2×C5 | C22×C16 |
| Frattini | C16 | C22×D5 |
| Fitting | C2×C5×C32 | C2 |
| prime radical | #709 | C8 |
| cosocle | C5×C16 | C23 |
| hypercenter | C2×C32 | D5 |
| hypocenter | C5 | #2258 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C32 ◁ C2×C32 | |
106 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=d,d^2=e,e^2=f,f^2=g,c^b=cg,h^b=h^4208 conjugacy classes — view the full character table.
| maximal subgroup of | #11987112, #11987114, #12019862, … |
| maximal normal subgroup of | #11987112, #11987114, #12019862, … |
| maximal quotient of | #12019478, #12019479, #12019495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2241);