Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C5 ⋊ #2923 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D5 |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C5×#408 | C2 |
| prime radical | C2×C5×#41 | C22 |
| cosocle | C2×C4×C5 | C24 |
| hypercenter | #408 | D5 |
| hypocenter | C5 | #2923 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ #408 | |
384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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,c^2=e,f^2=g,b^a=be,c^a=cg,d^b=dg,h^c=h^4100 conjugacy classes — view the full character table.
| maximal subgroup of | #12008737, #12015582, #12015583, … |
| maximal normal subgroup of | #12008737, #12015582, #12015583, … |
| maximal quotient of | #12096788, #12096837, #12097193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 12462);