Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #340 |
| prime radical | C23 | #103 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
104 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)(9,10,13,19,16,21,15,32)(11,22,24,34,27,36,26,40)(12,28,30,37,31,20,14,18)(17,25,33,29,35,39,38,23);(1,3,6,2)(4,7,5,8)(9,11,14,25,15,26,31,23,16,27,30,39,13,24,12,29)(10,17,20,34,19,33,18,36,21,35,28,40,32,38,37,22)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7297, #127940, #127941, … |
| maximal normal subgroup of | C2×#7297, #127940, #127941, … |
| maximal quotient of | #93406, #93419, #93451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 285);