Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18,15,19,13,16)(11,12,20,23,21,22,24,17);(1,3,4,7)(2,5,6,8)(10,16,19,18)(11,20,21,24)(12,22)(17,23);(2,6)(5,8)(9,11)(10,17)(12,16)(13,20)(14,24)(15,21)(18,22)(19,23);(9,12,14,23,15,22,13,17)(10,11,16,24,19,21,18,20)Polycyclic presentation:
2^8|a^2=g,b^2=ef,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27812, #6457707, #6457708, … |
| maximal normal subgroup of | C2×#27812, #6457707, #6457708, … |
| maximal quotient of | #315087, #315088, #315091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20800);