Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#456 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6,10,8)(7,13,16,20)(9,17,19,12)(11,18,14,15);(5,7)(6,11)(8,14)(9,18)(10,16)(12,20)(13,17)(15,19);(5,8,10,6)(7,14,16,11)(9,12,19,17)(13,18,20,15)64 conjugacy classes — view the full character table.
| maximal subgroup of | #136826, #136828, #136854, … |
| maximal normal subgroup of | #136826, #136828, #136854, … |
| maximal quotient of | C8×#456, #208751, #208759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14460);