Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,16)(12,22)(13,23)(14,24)(15,19)(17,20)(18,21);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,16,19,24)(12,20,13,21)(17,23,18,22);(6,8)(9,12,15,13)(10,17,19,18)(11,20,14,21)(16,23,24,22);(2,4)(9,13,15,12)(10,18,19,17)(11,21,14,20)(16,22,24,23)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,f^c=fh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6632013, #6632017, #6632023, … |
| maximal normal subgroup of | #6632013, #6632017, #6632023, … |
| maximal quotient of | #332903, #332904, #332905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23496);