Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #382 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10)(11,19,22,15)(12,23,25,38)(13,18)(14,17)(16,24,29,39)(20,33)(21,32,36,28)(26,30)(27,34)(31,40)(35,37);(1,2,5,3,6,8,4,7)(9,11,13,21,14,22,26,36)(10,15,18,28,17,19,30,32)(12,20,24,34,25,35,39,40)(16,27,23,37,29,31,38,33);(1,3,4,2,6,7,5,8)(9,12)(10,16,17,29)(11,20)(13,24)(14,25)(15,27,19,31)(18,23,30,38)(21,34)(22,35)(26,39)(28,37,32,33)(36,40)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fgh,f^c=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #134985, #134986, #134996, … |
| maximal normal subgroup of | #134985, #134986, #134996, … |
| maximal quotient of | #99276, #99280, #99283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1778);