Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21)(26,27)(30,31);(2,5)(6,8)(10,13)(14,15)(25,26,29,27)(28,30,32,31);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,19,20,24)(18,21,22,23)(25,27,29,26)(28,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^8|a^2=efg,b^2=fh,c^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16717, #611461, #611539, … |
| maximal normal subgroup of | C2×#16717, #611461, #611539, … |
| maximal quotient of | #155689, #155719, #155811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9705);