Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2179 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×D8 |
| commutator | C8 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C4×D8 |
| prime radical | C7×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×C8 ◁ C4×C7×C8 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,20,30,19)(10,21,23,36,25,38,24,32)(11,17,26,35,27,16,12,28)(15,31,33,39,34,29,37,22);(8,10)(9,15)(11,22)(12,29)(13,23)(14,25)(16,32)(17,36)(18,33)(19,37)(20,34)(21,28)(24,30)(26,31)(27,39)(35,38);(8,11,13,26,14,27,30,12)(9,16,18,28,20,17,19,35)(10,22,23,31,25,39,24,29)(15,32,33,21,34,36,37,38)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5414);