Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #2183 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D5 |
| commutator, socle, prime radical | C22×C5 | C22×C8 |
| Fitting | C5×#346 | C2 |
| cosocle | C2×C5×C8 | C23 |
| hypercenter | #346 | D5 |
| hypocenter | C5 | #2183 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C8 ◁ #346 | |
126 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=d,c^2=de,e^2=f,f^2=g,b^a=bd,c^a=cg,h^b=h^4136 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2183, C2×#10599861, #12252791, … |
| maximal normal subgroup of | D5×#2183, C2×#10599861, #12252791, … |
| maximal quotient of | #12014648, #12014652, #12014656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 1943);