Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2046 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C5 |
| prime radical | C22×C5×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
276 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,18)(12,19,20,16)(13,15,22,17)(21,24,25,23);(11,15)(12,20)(13,22)(16,24)(17,18)(19,23);(7,9)(10,12,13,21,14,20,22,25)(11,16,17,24,18,19,15,23)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5636);