Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#1331 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | D8 |
| commutator | C8 | C2×C7×C8 |
| Frattini, cosocle | C4×C8 | C22×C7 |
| socle | C22×C7 | #102 |
| prime radical | C2×C7×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C42×C7 ◁ C4×C7×C8 ◁ G | |
110 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(16,17)(18,21)(19,28)(20,24)(22,25)(23,26)(27,30)(29,31);(17,21)(18,25)(19,26)(22,23)(24,30)(27,31)(28,29)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 463);