Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6)(3,7,8,5)(10,15)(11,17)(12,20)(13,23)(16,25)(18,34)(19,35)(21,37)(22,29)(24,36)(26,27)(28,31)(30,40)(32,33)(38,39);(1,3,4,8)(2,5,6,7)(10,16)(15,25)(18,30)(19,31)(26,38)(27,39)(28,35)(34,40);(9,10,17,27,12,18,32,35,23,31,37,40,36,38,22,25,14,16,29,39,24,30,21,28,13,19,33,34,20,26,11,15)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ch46 conjugacy classes — view the full character table.
| maximal subgroup of | #151602, #151603, #151604, … |
| maximal normal subgroup of | #151602, #151603, #151604, … |
| maximal quotient of | #152404, #152853, #152854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6722);