Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#389 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 16 points:
(3,4);(1,2)(3,4);(6,8)(10,12)(13,14,16,15);(9,10)(11,12)(13,15,16,14);(5,6,7,8)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#389 |
| maximal subgroup of | #96911, #97077, #106137, … |
| maximal normal subgroup of | #96911, #97077, #106137, … |
| maximal quotient of | C22×C4⋊#95, C22×#1436, C2×C4×#389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27006);