Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2416 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4)(10,13)(11,15)(14,16);(5,6)(7,8)(9,10,11,13,12,14,15,16);(2,4)(6,8)(10,14)(13,16);(9,10,15,16,12,14,11,13)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24794, #6442895, #6442903, … |
| maximal normal subgroup of | C2×#24794, #6442895, #6442903, … |
| maximal quotient of | #277661, #277665, #277669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17782);