Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C5×C4⋊Q8 ⋊ C4 |
| Sylow | C5 ⋊ #2369 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C5×C4⋊Q8 | C4 |
| prime radical, cosocle | C22×D5 | C24 |
| hypercenter | C4⋊Q8 | AGL(1,𝔽5) |
| hypocenter | C5 | #2369 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
496 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 188 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|a^2=g,b^2=f,c^2=fg,d^2=e,b^a=be,c^a=cfg,h^a=h^2,d^b=de,h^b=h^3,h^f=h^4,h^g=h^464 conjugacy classes — view the full character table.
| maximal subgroup of | #13019063, #13019064, #13019065, … |
| maximal normal subgroup of | #13019063, #13019064, #13019065, … |
| maximal quotient of | #13028006, #13028033, #13028048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19471);