Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C7×#2183 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C23 |
| commutator | C22 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle, prime radical | C22×C7 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,15)(11,13)(12,14)(16,17,19,23,20,18,25,26,21,24,28,31,29,27,22,30);(8,10,12,11)(9,13,14,15);(8,11,12,10)(9,13,14,15)(16,18,19,26,20,24,25,31,21,27,28,30,29,17,22,23)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5418);