Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#407 ⋊ C2 |
| Sylow | C5 ⋊ #3194 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D20 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#407 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #407 | D5 |
| hypocenter | C5 | #3194 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #407 | |
466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|b^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^4,c^b=ceg,f^c=fg,h^c=h^482 conjugacy classes — view the full character table.
| maximal subgroup of | #11996257, #11996259, #12073456, … |
| maximal normal subgroup of | #11996257, #11996259, #12073456, … |
| maximal quotient of | #12108375, #12108391, #12109592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 12456);