Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×#1425 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | D16 |
| commutator | C16 | C2×C4×C7 |
| Frattini, cosocle | C2×C16 | C22×C7 |
| socle | C2×C7 | #366 |
| prime radical | C22×C7 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ C2×C7×C16 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,12,18,15,20,14,36)(10,21,24,46,27,48,26,58)(11,28,30,59,32,61,31,62)(13,34,33,44,35,63,29,19)(16,37,38,54,40,53,64,23)(17,41,42,51,43,50,60,22)(25,56,55,68,57,71,52,47)(39,49,45,65,66,70,67,69);(8,10,22,49,13,25,30,37,12,24,41,65,33,55,32,54,15,27,51,70,35,57,31,53,14,26,50,69,29,52,11,23)(9,16,28,47,19,39,60,58,36,64,62,71,63,67,43,48,20,40,61,68,44,66,42,46,18,38,59,56,34,45,17,21)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 557);