Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#336 ⋊ C2 |
| Sylow | C5 ⋊ #2011 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D20 |
| commutator, cosocle | C22×C4×C5 | C23 |
| Frattini | C22×C4 | C22×D5 |
| Fitting | C5×#336 | C2 |
| socle | C23×C5 | C2×D4 |
| hypercenter | #336 | D5 |
| hypocenter | C5 | #2011 |
| LCS | G ▷ C22×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #336 | |
510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 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|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^4,c^b=cf,d^b=dg,e^b=eg,h^b=h^4,d^c=dg,e^c=eg,h^c=h^4112 conjugacy classes — view the full character table.
| maximal subgroup of | #11987999, #11988004, #11988006, … |
| maximal normal subgroup of | #11987999, #11988004, #11988006, … |
| maximal quotient of | #11999723, #11999765, #11999833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 1549);