Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C5 ⋊ #1446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D5 |
| commutator | C23×C5 | C22×C4 |
| Fitting | C2×C5×#95 | C2 |
| socle, prime radical, cosocle | C24×C5 | C23 |
| hypercenter | C2×#95 | D5 |
| hypocenter | C5 | #1446 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C24 ◁ C2×#95 | |
736 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 209 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=g,b^2=f,c^2=f,b^a=bd,c^a=ce,c^b=cf,h^b=h^4124 conjugacy classes — view the full character table.
| maximal subgroup of | #11986117, #11991435, #11994421, … |
| maximal normal subgroup of | #11986117, #11991435, #11994421, … |
| maximal quotient of | #11987226, #11987228, #12008206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2299);