Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3552 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,13)(15,20)(16,17)(18,26)(28,29);(7,11)(8,12)(14,15,19,20)(16,22,17,21)(18,23,26,27)(24,29,25,28);(14,16,19,17)(15,21,20,22)(18,24,26,25)(23,28,27,29);(6,8,10,12)(7,9,11,13)(14,17,19,16)(15,22,20,21)(18,25,26,24)(23,29,27,28);(6,9)(7,12)(8,11)(10,13)(14,18)(15,23)(16,24)(17,25)(19,26)(20,27)(21,28)(22,29)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21419);