Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#1385 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #106 |
| commutator | C8 | C42×C7 |
| Frattini, cosocle | C22×C8 | C22×C7 |
| socle | C2×C7 | #340 |
| prime radical | C23×C7 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
124 conjugacy classes of subgroups.
| 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)(8,9,12,18,15,20,14,30)(10,21,23,36,25,37,28,32)(11,19,13,29,27,35,26,17)(16,24,31,22,34,38,33,39);(8,10,13,24,12,28,27,39,15,25,26,38,14,23,11,22)(9,16,19,32,30,31,17,21,20,34,35,36,18,33,29,37)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 517);