Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#1389 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #102 |
| commutator | C8 | C42×C7 |
| Frattini, cosocle | C22×C8 | C22×C7 |
| socle | C2×C7 | #340 |
| prime radical | C23×C7 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,14,18,13,29)(10,19,21,36,24,32,23,38)(11,25,26,34,28,39,27,16)(15,30,31,22,33,37,35,20);(8,10,11,22,12,21,26,37,14,24,28,20,13,23,27,30)(9,15,16,32,29,35,39,36,18,33,34,19,17,31,25,38)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 521);