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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
864 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)(9,10,12,14)(11,13,15,16)(18,19)(22,23);(1,3,6,5)(2,7,4,8)(17,18,21,19)(20,22,24,23);(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22);(1,4,6,2)(3,8,5,7)(9,11,12,15)(10,13,14,16)(17,20,21,24)(18,22,19,23);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^8|a^2=fg,b^2=f,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 | #6579667, #6579679, #6579681, … |
| maximal normal subgroup of | #6579667, #6579679, #6579681, … |
| maximal quotient of | #517727, #517733, #517750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28821);