Finite Group DatabasebetaFinite Group Databasebeta| order | 992 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C31 · Schur multiplier (H2) = C1 |
| direct product | C31×Q32 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C31 | D8 |
| commutator, Frattini, cosocle | C8 | C22×C31 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C4×C31 ◁ C8×C31 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 2 generators on 63 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)(48,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49);(32,48,40,56)(33,49,41,57)(34,50,42,58)(35,51,43,59)(36,52,44,60)(37,53,45,61)(38,54,46,62)(39,55,47,63)341 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(992, 62);