Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #1917 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D5 |
| commutator, socle, prime radical | C23×C5 | C22×C4 |
| Fitting | C5×#336 | C2 |
| cosocle | C22×C4×C5 | C23 |
| hypercenter | #336 | D5 |
| hypocenter | C5 | #1917 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C22×C4 ◁ #336 | |
228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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=f,b^2=f,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^4,c^b=cg,h^c=h^4124 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1917, C2×#10599405, #12101362, … |
| maximal normal subgroup of | D5×#1917, C2×#10599405, #12101362, … |
| maximal quotient of | #11997446, #11997453, #11997460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 1487);