Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C4×#525 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(6,8);(9,11,12,15)(10,14,13,16)112 conjugacy classes — view the full character table.
| maximal subgroup of | #106053, #111559, #129369, … |
| maximal normal subgroup of | #106053, #111559, #129369, … |
| maximal quotient of | C4×#2290, C4×#2297, C4×#2296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27019);