Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3204 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ G | |
402 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)(8,12)(9,11)(10,13)(14,15,18,23)(16,25,27,20)(17,24,19,22)(21,28,29,26);(6,8)(7,10)(9,13)(11,12)(14,16,18,27)(15,20,23,25)(17,28,19,26)(21,22,29,24);(8,13)(10,12)(14,16)(15,21)(17,26)(18,27)(19,28)(20,24)(22,25)(23,29);(7,11)(10,12)(14,17,18,19)(15,22,23,24)(16,26,27,28)(20,21,25,29)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18881);