Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator, socle, prime radical | C23 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
219 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6,9,11,8,10,12,7)(13,14,16,15,17,18,20,19);(2,4)(6,10)(7,11)(13,14,16,15,17,18,20,19);(5,7,12,10,8,11,9,6)(13,15,16,18,17,19,20,14)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,b^a=bd,c^a=ch,c^b=cg88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8265, #183289, #183290, … |
| maximal normal subgroup of | C2×#8265, #183289, #183290, … |
| maximal quotient of | #96904, #96905, #96908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1253);