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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,7,8)(3,4,6,5)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,22,20,23);(1,3,7,6)(2,5,8,4)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24);(1,4,7,5)(2,3,8,6)(9,10,13,12)(11,15,16,14)(17,19,21,20)(18,22,24,23);(1,5,7,4)(2,3,8,6)(9,11)(10,15)(12,14)(13,16)(17,20,21,19)(18,23,24,22);(1,6,7,3)(2,5,8,4)(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=fgh,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 | #6586335, #6586338, #6586343, … |
| maximal normal subgroup of | #6586335, #6586338, #6586343, … |
| maximal quotient of | #517872, #517878, #517906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28859);