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×#1479 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D4 |
| commutator | C4 | C22×C7×C8 |
| Frattini, cosocle | C42 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C4 |
| prime radical | C23×C7 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
248 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(16,17)(18,25)(19,23)(20,29)(21,24)(22,27)(26,30)(28,31);(16,18,21,28)(17,22,24,30)(19,27,20,26)(23,31,29,25);(16,19,21,20)(17,23,24,29)(18,26,28,27)(22,25,30,31)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4714);