Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×Q8⋊C8 ⋊ C2 |
| Sylow | C5 ⋊ #1621 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4028 |
| commutator, cosocle | C42×C5 | C23 |
| Frattini | C42 | C22×D5 |
| Fitting | C5×Q8⋊C8 | C2 |
| socle | C22×C5 | #120 |
| hypercenter | Q8⋊C8 | D5 |
| hypocenter | C5 | #1621 |
| LCS | G ▷ C42×C5 ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ Q8⋊C8 | |
278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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|c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^4,d^b=df,e^b=eg,h^b=h^4,d^c=dg,e^c=eg82 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1621, C2×#10599052, #12075199, … |
| maximal normal subgroup of | D5×#1621, C2×#10599052, #12075199, … |
| maximal quotient of | #11988004, #11988016, #11988035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 1134);