Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#2923 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C24 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C23×C4 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,17,21,18,20,24,27)(16,19,23,26,22,25,28,29);(7,10)(11,13)(14,16)(15,19)(17,23)(18,22)(20,25)(21,26)(24,28)(27,29);(6,8,9,12)(7,11,10,13)(15,20)(19,25)(21,27)(26,29);(16,22)(19,25)(23,28)(26,29)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18600);