Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16);(10,13)(14,15);(1,2)(3,4)(5,6)(7,8);(9,10)(11,14)(12,13)(15,16);(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^8|a^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| Fitting subgroup of | #34175⋊C3 |
| maximal subgroup of | #105862, #106878, #107000, … |
| maximal normal subgroup of | #105862, #106878, #107000, … |
| maximal quotient of | #513549, #513680, #513714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27163);