Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator, socle | C23 | C2×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
309 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 131 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)(5,6,8,7,9,10,12,11);(2,4)(13,14,15,18,16,17,19,20);(5,7,8,10,9,11,12,6)(14,17)(18,20)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,b^a=bd,c^a=cg,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #93545, #124776, #124839, … |
| maximal normal subgroup of | #93545, #124776, #124839, … |
| maximal quotient of | #96678, #96679, #96704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1135);