Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#2913 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator | C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ G | |
476 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 344 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) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(11,15)(13,16)(17,21)(18,22)(20,23)(24,25);(10,11,16,22,14,18,13,15)(12,17,23,25,19,24,20,21);(10,12)(11,17)(13,20)(14,19)(15,21)(16,23)(18,24)(22,25);(12,19)(17,24)(20,23)(21,25)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18590);