Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#1785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
584 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 136 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)(10,11,14,19,15,18,13,17)(12,20,21,25,22,16,23,24);(10,12)(11,16)(13,21)(14,23)(15,22)(17,24)(18,20)(19,25);(7,9)(10,13,15,14)(11,17,18,19)(12,21,22,23)(16,24,20,25)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5375);