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×#1386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #102 |
| commutator | C8 | C42×C7 |
| Frattini, cosocle | C22×C8 | C22×C7 |
| prime radical | C23×C7 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
140 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13,16,20)(14,15,22,24)(17,19,25,27)(18,26,23,21);(12,14,17,21,18,19,24,20,16,22,25,26,23,27,15,13)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 518);