Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,5,7)(11,15)(14,16)(17,18,22,21)(19,24,23,20);(1,3,6,5)(2,7,4,8)(9,10,13,12)(11,14,15,16)(17,19)(18,20)(21,24)(22,23);(1,2,6,4)(3,8,5,7)(9,11)(10,14)(12,16)(13,15)(17,20,22,24)(18,23,21,19);(1,4,6,2)(3,8,5,7)(9,12,13,10)(11,16,15,14)(17,21,22,18)(19,24,23,20);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,d^2=gh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6580777, #6580780, #6580840, … |
| maximal normal subgroup of | #6580777, #6580780, #6580840, … |
| maximal quotient of | #517943, #517948, #517961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28888);