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 | #2407 | 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)(9,10,13,18,15,20,14,19)(11,16,21,24,22,17,12,23);(2,4)(5,6)(7,8)(9,11,13,21,15,22,14,12)(10,16,18,24,20,17,19,23);(6,8)(9,12,15,21)(10,17,20,16)(11,13,22,14)(18,24,19,23);(2,4)(9,13,15,14)(10,18,20,19)(11,12,22,21)(16,23,17,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25119, #6438933, #6438952, … |
| maximal normal subgroup of | C2×#25119, #6438933, #6438952, … |
| maximal quotient of | #281792, #281794, #281800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18107);