Finite Group DatabasebetaFinite Group Databasebeta| order | 992 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C25 |
| direct product | C2×#41×C31 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C31 |
| socle | C22×C31 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ G | |
164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 496 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(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);(3,4)(5,8)(6,7)(9,10);(4,7)(8,10);(3,5,6,9)(4,8,7,10)620 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(992, 191);