Finite Group DatabasebetaFinite Group Databasebeta| order | 992 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C31 | C22 |
| commutator | C2 | C22×C4×C31 |
| Frattini, cosocle | C4 | C23×C31 |
| socle | C2×C31 | C22×C4 |
| prime radical | #41×C31 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C31 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 496 |
A minimal faithful permutation representation — 3 generators on 47 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)(32,33,35,38,36,39,42,45)(34,37,41,44,40,43,46,47);(32,34)(33,37)(35,41)(36,40)(38,44)(39,43)(42,46)(45,47);(34,40)(37,43)(41,46)(44,47)620 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(992, 166);