Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#331 ⋊ C2 |
| Sylow | C5 ⋊ #1619 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4028 |
| commutator, cosocle | C42×C5 | C23 |
| Frattini | C42 | C22×D5 |
| Fitting | C5×#331 | C2 |
| socle | C22×C5 | #120 |
| hypercenter | #331 | D5 |
| hypocenter | C5 | #1619 |
| LCS | G ▷ C42×C5 ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #331 | |
336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 336 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) = 160 |
Polycyclic presentation:
2^7,5|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,d^c=dg,e^c=eg,h^c=h^464 conjugacy classes — view the full character table.
| maximal subgroup of | #12020012, D5×#1619, C2×#10599183, … |
| maximal normal subgroup of | #12020012, D5×#1619, C2×#10599183, … |
| maximal quotient of | #11987999, #11988018, #11988023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 1265);