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) = C22 |
| direct product | C7×#2153 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C2×D4 |
| commutator | C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C2×C4⋊C8 |
| prime radical | C23×C7 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
152 conjugacy classes of subgroups.
| 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)(8,9,12,17,13,16,26,31,14,18,27,32,11,24,25,37)(10,19,22,34,23,29,36,38,20,33,35,39,21,15,28,30);(8,10,11,21,14,20,13,23)(9,15,16,19,18,29,24,33)(12,22,25,28,27,35,26,36)(17,30,31,34,32,38,37,39);(10,20)(15,29)(19,33)(21,23)(22,35)(28,36)(30,38)(34,39)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5388);