Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C2×C5×H(5,𝔽2) ⋊ C2 |
| Sylow | C5 ⋊ H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D5 |
| commutator, socle, cosocle | C2×C5 | C26 |
| Fitting | C2×C5×H(5,𝔽2) | C2 |
| hypercenter | C2×H(5,𝔽2) | D5 |
| hypocenter | C5 | H(7,𝔽2) |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×H(5,𝔽2) | |
5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^7,5|b^a=bg,f^a=fg,h^a=h^4,e^b=eg,d^c=dg133 conjugacy classes — view the full character table.
| maximal subgroup of | #12248042, #12248046, #12248056, … |
| maximal normal subgroup of | #12248042, #12248046, #12248056, … |
| maximal quotient of | #13005916, #13005917, #13005919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21518);