Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#408 ⋊ C2 |
| Sylow | C5 ⋊ #2018 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4076 |
| commutator, cosocle | C22×C4×C5 | C23 |
| Frattini | C22×C4 | C22×D5 |
| Fitting | C5×#408 | C2 |
| socle | C22×C5 | #121 |
| hypercenter | #408 | D5 |
| hypocenter | C5 | #2018 |
| LCS | G ▷ C22×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #408 | |
382 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 73 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|d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^4,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eg,f^c=fg,h^c=h^482 conjugacy classes — view the full character table.
| maximal subgroup of | #11986794, #11986796, #11986800, … |
| maximal normal subgroup of | #11986794, #11986796, #11986800, … |
| maximal quotient of | #11999900, #11999917, #11999967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2881);