Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×D16 | C22 |
| 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 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,19)(11,15,23,30)(12,24,26,40)(13,27,25,18)(16,31,33,20)(17,34,35,21)(22,36,37,29)(28,39,32,38);(1,3,4,8)(2,5,6,7);(10,15)(11,20)(12,21)(13,25)(16,17)(18,36)(19,30)(22,39)(23,31)(24,28)(26,34)(27,29)(32,40)(33,35)(37,38)Polycyclic presentation:
2^8|a^2=dh,b^2=d,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #151570, #151573, #151576, … |
| maximal normal subgroup of | #151570, #151573, #151576, … |
| maximal quotient of | #152269, #152435, #152797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6693);