Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C8 |
No aliases for this group.
| Sylow | C5 ⋊ #1453 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D5 |
| commutator, socle | C22×C5 | C2×C42 |
| Fitting | C5×#431 | C2 |
| prime radical | C23×C5 | C42 |
| cosocle | C42×C5 | C23 |
| hypercenter | #431 | D5 |
| hypocenter | C5 | #1453 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C42 ◁ #431 | |
190 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 105 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=d,b^2=e,d^2=f,e^2=g,c^a=cg,h^a=h^4,c^b=cf136 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1453, C2×#10601536, #12064932, … |
| maximal normal subgroup of | D5×#1453, C2×#10601536, #12064932, … |
| maximal quotient of | #11988668, #11988688, #11989732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 3618);