Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×C8×Q8 ⋊ C2 |
| Sylow | C5 ⋊ #2991 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D5 |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C5×C8×Q8 | C2 |
| prime radical | C2×#713 | C4 |
| cosocle | C2×C4×C5 | C24 |
| hypercenter | C8×Q8 | D5 |
| hypocenter | C5 | #2991 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C8×Q8 | |
394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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=f,d^2=e,f^2=g,b^a=be,c^a=cg,d^a=de,c^b=ceg,h^b=h^4,h^c=h^4,h^d=h^4130 conjugacy classes — view the full character table.
| maximal subgroup of | #12070501, #12070507, #12070512, … |
| maximal normal subgroup of | #12070501, #12070507, #12070512, … |
| maximal quotient of | #12098139, #12098325, #12098362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 13720);