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×#3554 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
1022 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)(8,12)(9,10)(11,13)(15,19)(16,22)(17,25)(27,29);(7,10)(11,12)(14,15,18,19)(16,23,22,20)(17,21,25,26)(24,29,28,27);(14,16,18,22)(15,20,19,23)(17,24,25,28)(21,27,26,29);(6,7)(8,11)(9,10)(12,13)(14,15)(16,20)(17,21)(18,19)(22,23)(24,27)(25,26)(28,29);(6,8,9,13)(7,11,10,12)(14,17,18,25)(15,21,19,26)(16,24,22,28)(20,27,23,29)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21421);