Finite Group DatabasebetaFinite Group Databasebeta| order | 999 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C4×C92 (order 648) · Inn(G) = C1 · Out(G) = C2×C4×C92 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C999 |
| abelian | C27×C37 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C9 | C3×C37 |
| socle, prime radical | C3×C37 | C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 999 |
A minimal faithful permutation representation — 1 generator on 64 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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)999 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(999, 2);