Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#1323 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22⋊C4 |
| commutator | C2×C4 | C2×C5×C8 |
| Frattini, cosocle | C22×C8 | C22×C5 |
| socle | C22×C5 | C22⋊C8 |
| prime radical | C23×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,18,23,19,24,33,41,20,25,34,42,17,22,31,39)(16,26,28,43,27,37,44,40,29,38,32,45,30,21,35,36);(6,8,9,13)(7,10,11,12)(14,16)(15,21)(17,30)(18,32)(19,27)(20,29)(22,38)(23,40)(24,26)(25,37)(28,34)(31,44)(33,35)(36,42)(39,43)(41,45)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 456);