Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #340 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #103 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,5,7,6,3,4,8)(9,10,13,19,16,21,15,31)(11,22,24,37,26,38,29,33)(12,20,14,30,28,36,27,18)(17,25,32,23,35,39,34,40);(1,3,6,2)(4,7,5,8)(9,11,14,25,13,29,28,40,16,26,27,39,15,24,12,23)(10,17,20,33,31,32,18,22,21,35,36,37,19,34,30,38)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=gh,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7303, #127964, #127965, … |
| maximal normal subgroup of | C2×#7303, #127964, #127965, … |
| maximal quotient of | #93409, #93421, #93453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 291);