Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 196608 |
| 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 | |
790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,7,8)(3,4,6,5)(11,15)(13,16)(17,18,20,22)(19,21,23,24);(1,3,7,6)(2,5,8,4)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,22)(23,24);(1,4,7,5)(2,3,8,6)(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,21,22,24);(1,5,7,4)(2,3,8,6);(1,6,7,3)(2,5,8,4)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #35868⋊C3 |
| maximal subgroup of | #6579477, #6579480, #6579481, … |
| maximal normal subgroup of | #6579477, #6579480, #6579481, … |
| maximal quotient of | #517872, #517900, #517915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28856);