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×#3547 |
| 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)(7,10)(8,12)(14,15,18,19)(16,22,23,20)(17,25,26,21)(24,27,29,28);(7,10)(11,13)(15,19)(16,23)(21,25)(24,29);(6,7,9,10)(8,11,12,13)(14,16,18,23)(15,20,19,22)(17,24,26,29)(21,27,25,28);(6,8)(7,11)(9,12)(10,13)(14,15)(16,20)(17,21)(18,19)(22,23)(24,27)(25,26)(28,29);(14,17,18,26)(15,21,19,25)(16,24,23,29)(20,27,22,28)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21414);