Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#500 ⋊ C2 |
| Sylow | C5 ⋊ #3169 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#500 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #500 | D5 |
| hypocenter | C5 | #3169 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #500 | |
456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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|c^2=fg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^4,h^b=h^4,h^d=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #12010624, #12010629, #12010644, … |
| maximal normal subgroup of | #12010624, #12010629, #12010644, … |
| maximal quotient of | #12099837, #12103513, #12104481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15746);