Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C4×#520 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C2 | C23×C42 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(5,6,7,8)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16);(10,13)(14,16);(9,11,12,15)(10,14,13,16)136 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×#520⋊C3 |
| maximal subgroup of | #105908, #106408, #107778, … |
| maximal normal subgroup of | #105908, #106408, #107778, … |
| maximal quotient of | C4×#2275, C4×#2276, C4×#2274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26989);