Finite Group DatabasebetaFinite Group Databasebeta| order | 992 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C31 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 248 |
A minimal faithful permutation representation — 4 generators on 39 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)(34,38)(35,36)(37,39);(33,36)(37,38);(32,33)(34,37)(35,36)(38,39);(32,34)(33,37)(35,39)(36,38)527 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(992, 192);