Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C2×D4 · Out(G) = C210 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#478 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6,8,10)(7,11,12,9)(13,14,15,17)(16,18,20,19);(5,7,8,12)(6,9,10,11)(14,17)(18,19);(14,18)(16,20)(17,19)76 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#478, #212310, #212316, … |
| maximal normal subgroup of | C8×#478, #212310, #212316, … |
| maximal quotient of | C8×#478, #212316, #212344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14599);