Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C2×C5×#536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C22×C5×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(23,24,25,26,27);(4,6)(8,12)(9,15)(10,18)(20,22);(7,8,11,12)(9,16,15,13)(10,14,18,19)(17,22,21,20);(3,4)(5,6)(7,9,11,15)(8,13,12,16)(10,17,18,21)(14,20,19,22);(7,10)(8,14)(9,17)(11,18)(12,19)(13,20)(15,21)(16,22)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21314);