Finite Group DatabasebetaFinite Group Databasebeta| order | 992 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C31 |
| socle | C2×C31 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C2×C4×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)(34,41)(35,39)(36,44)(37,40)(38,42)(43,46)(45,47);(32,34)(33,38)(35,42)(36,45)(37,43)(39,46)(40,47)(41,44);(32,35,37,36)(33,39,40,44)(34,42,43,45)(38,46,47,41)434 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(992, 170);