Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C7×#361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #102 |
| commutator | C8 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C22×C7×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
256 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(19,20,21,22,23,24,25);(3,4,6,10,7,12,17,11)(5,13,15,9,16,8,14,18);(4,8)(5,14)(6,17)(9,10)(11,18)(12,13)(15,16)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5381);