Finite Group DatabasebetaFinite Group Databasebeta| order | 645 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×C4×C7 (order 336) · Inn(G) = C1 · Out(G) = C22×C3×C4×C7 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C645 |
| abelian | C3×C5×C43 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 645 |
A minimal faithful permutation representation — 1 generator on 51 points:
(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)645 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(645, 2);