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) = C23 |
| direct product | C7×#2117 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C2×D4 |
| commutator | C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C2×C22⋊C8 |
| prime radical | C2×C7×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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)(8,9,12,18,13,19,27,36,14,16,28,32,11,17,25,34)(10,20,22,37,21,30,38,35,23,33,26,39,24,15,29,31);(8,10)(9,15)(11,24)(12,26)(13,21)(14,23)(16,30)(17,33)(18,35)(19,20)(22,28)(25,38)(27,29)(31,32)(34,37)(36,39);(9,16)(15,30)(17,19)(18,32)(20,33)(31,35)(34,36)(37,39)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5352);