Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#2171 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C2×D4 |
| commutator | C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C8×D4 |
| prime radical | C7×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×C16 ◁ G | |
144 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(8,9,13,19,14,18,31,37,15,20,32,22,12,28,30,38)(10,21,24,26,23,16,33,27,17,34,36,25,29,35,39,11);(8,10)(9,16)(11,22)(12,29)(13,24)(14,23)(15,17)(18,34)(19,27)(20,35)(21,28)(25,37)(26,38)(30,39)(31,33)(32,36);(8,11,13,21,14,26,31,16,15,27,32,34,12,25,30,35)(9,17,19,36,18,29,37,39,20,10,22,24,28,23,38,33)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5406);