Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8534 |
| automorphism | Aut(G) = #72285069 (order 4096) · Inn(G) = #379 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C25 | C23 |
| prime radical | #3490 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
531 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(5,8)(9,10)(11,15,16,14)(12,13);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16);(1,3)(2,5)(4,7)(6,8)(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^8|a^2=f,c^2=g,b^a=bd,c^a=ce,d^a=dg,e^b=eh,f^b=fg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94453, #94530, #140024, … |
| maximal normal subgroup of | #94453, #94530, #140024, … |
| maximal quotient of | #97054, #97056, #97057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1523);