Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #2987 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D5 |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C5×#533 | C2 |
| prime radical | C23×C5 | C22×C4 |
| cosocle | C2×C4×C5 | C24 |
| hypercenter | #533 | D5 |
| hypocenter | C5 | #2987 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
314 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 197 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=f,b^2=e,c^2=e,f^2=g,b^a=be,c^a=cg,d^a=de,h^a=h^4,c^b=ceg,d^b=dg94 conjugacy classes — view the full character table.
| maximal subgroup of | #11993707, #11993710, #11994252, … |
| maximal normal subgroup of | #11993707, #11993710, #11994252, … |
| maximal quotient of | #12098064, #12098085, #12098400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 17146);