Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#394 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1275 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(14,15);(9,10)(11,12)(13,14)(15,16);(6,8)(10,12)(13,15,16,14)88 conjugacy classes — view the full character table.
| maximal subgroup of | #99552, #99553, #99568, … |
| maximal normal subgroup of | #99552, #99553, #99568, … |
| maximal quotient of | C4×#1438, C4×#1440, C4×#1439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6846);