Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)(25,26,28,33,29,32,37,31)(27,30,35,39,36,38,34,40);(1,3,4,7)(2,5,6,8)(10,14)(15,21)(16,22)(23,24)(25,27,28,35,29,36,37,34)(26,30,33,39,32,38,31,40);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(26,31)(27,34)(28,37)(30,38)(32,33)(35,36);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(26,32)(30,38)(31,33)(39,40)Polycyclic presentation:
2^8|a^2=eg,b^2=f,c^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30232, #6436337, #6436342, … |
| maximal normal subgroup of | C2×#30232, #6436337, #6436342, … |
| maximal quotient of | #380214, #380215, #380216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23220);