Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator | C22 | C2×C4×C8 |
| socle | C23 | C2×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
289 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,8,11,9,7,10,12)(13,14,17,15,16,19,18,20);(2,4)(13,14,18,20,16,19,17,15);(5,7,8,12,9,6,10,11)(13,15,17,19,16,20,18,14)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,b^a=bd,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8148, #181892, #181894, … |
| maximal normal subgroup of | C2×#8148, #181892, #181894, … |
| maximal quotient of | #96680, #96681, #96708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1136);