Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C7×#1369 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #106 |
| commutator | C8 | C2×C7×C8 |
| Frattini, cosocle | C4×C8 | C22×C7 |
| socle | C2×C7 | #334 |
| prime radical | C22×C7 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×C8 ◁ C4×C7×C8 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,18,14,19,29,35,15,20,28,17,13,31,11,27)(10,21,23,32,25,36,22,33,26,34,30,39,24,38,37,16);(8,10,11,22,13,24,28,23,15,26,29,37,14,25,12,30)(9,16,17,21,19,32,27,36,20,33,18,34,31,39,35,38)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 501);