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 | #2215 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
93 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,26)(14,27)(17,23)(18,24)(19,22)(20,28)(21,38)(25,34)(29,35)(30,36)(31,39)(32,37)(33,40);(2,5)(6,8)(10,15)(16,28)(17,29)(18,30)(19,27)(26,38)(31,33)(32,34);(1,3,4,7)(2,6,5,8)(10,16)(12,23)(13,24)(14,22)(15,28)(17,18)(19,34)(20,35)(21,36)(25,40)(26,31)(27,32)(29,30)(33,38)(37,39)Polycyclic presentation:
2^8|c^2=dh,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 | #151483, #151485, #151486, … |
| maximal normal subgroup of | #151483, #151485, #151486, … |
| maximal quotient of | #152365, #152418, #152500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6700);