Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
579 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)(9,10,13,16,14,18,12,17)(11,15,21,23,22,24,20,19);(2,4)(5,6)(7,8)(9,11,12,20,14,22,13,21)(10,15,17,19,18,24,16,23);(6,8)(9,10,14,18)(11,19,22,23)(12,16,13,17)(15,21,24,20);(2,4)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25131, #6438924, #6438946, … |
| maximal normal subgroup of | C2×#25131, #6438924, #6438946, … |
| maximal quotient of | #282080, #282082, #282088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18119);