Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#487 ⋊ C2 |
| Sylow | C5 ⋊ #3303 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#487 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #487 | D5 |
| hypocenter | C5 | #3303 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #487 | |
398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=g,d^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,d^b=deg,h^b=h^4,f^c=fg,h^c=h^4,h^d=h^479 conjugacy classes — view the full character table.
| maximal subgroup of | #12088886, #12088893, #12088946, … |
| maximal normal subgroup of | #12088886, #12088893, #12088946, … |
| maximal quotient of | #12114757, #12115565, #12116462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15428);