Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(10,14)(15,17)(16,21)(22,23)(25,26,30,31)(27,34,32,28)(29,33,37,38)(35,40,39,36);(10,14)(12,20)(15,17)(18,24)(26,31)(27,32)(33,38)(35,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(1,4,5,3)(2,7,8,6)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,37)(26,33,31,38)(27,35,32,39)(28,36,34,40)Polycyclic presentation:
2^8|a^2=fgh,c^2=f,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41801, #7639806, #7640244, … |
| maximal normal subgroup of | C2×#41801, #7639806, #7640244, … |
| maximal quotient of | #544919, #545318, #545953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34789);