Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C5 ⋊ #2434 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C23×C5×Q8 | C2 |
| prime radical | C24×C5 | C23 |
| hypercenter | C23×Q8 | D5 |
| hypocenter | C5 | #2434 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
800 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 213 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=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg,h^c=h^4118 conjugacy classes — view the full character table.
| maximal subgroup of | #11997032, #11997034, #11999738, … |
| maximal normal subgroup of | #11997032, #11997034, #11999738, … |
| maximal quotient of | #12021281, #12021688, #12022169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 17878);