Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#1420 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #357 |
| commutator | C16 | C2×C4×C7 |
| Frattini, cosocle | C2×C16 | C22×C7 |
| prime radical | C22×C7 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ C2×C7×C16 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,12,18,15,20,14,38)(10,21,23,52,26,54,25,61)(11,27,30,62,33,64,32,69)(13,34,29,50,37,45,36,19)(16,39,40,22,42,56,48,58)(17,43,46,68,47,67,63,31)(24,59,55,44,60,71,35,53)(28,49,65,70,57,51,66,41);(8,10,15,26)(9,16,20,42)(11,28,33,57)(12,25,14,23)(13,35,37,55)(17,44,47,53)(18,48,38,40)(19,49,50,51)(21,27,54,64)(22,31,58,68)(24,29,60,36)(30,66,32,65)(34,41,45,70)(39,43,56,67)(46,59,63,71)(52,69,61,62)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 552);