Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,14,13)(11,16,12,15)(17,18,20,22)(19,23,24,21);(1,2)(3,8)(4,5)(6,7)(9,11,14,12)(10,15,13,16);(2,5)(6,8)(9,12,14,11)(10,15,13,16)(17,19,20,24)(18,21,22,23);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,15,12,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7118019, #7118021, #7118043, … |
| maximal normal subgroup of | #7118019, #7118021, #7118043, … |
| maximal quotient of | #535455, #535556, #536125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37942);