Finite Group DatabasebetaFinite Group Databasebeta| order | 992 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C23×C31 |
| prime radical | #41×C31 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ G | |
60 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)(34,40)(35,43)(36,38)(37,41)(39,45)(42,47)(44,46);(32,34,36,42)(33,37,38,46)(35,44,45,41)(39,47,43,40);(33,38)(37,46)(39,43)(40,47)341 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(992, 172);