Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
125 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)(3,8)(4,6)(5,7)(9,10,12,17,25,35,39,34,14,19,26,33,13,18,24,16)(11,15,21,28,22,29,38,40,23,31,37,27,20,30,36,32);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,21)(13,20)(14,23)(16,32)(17,28)(18,30)(19,31)(22,25)(24,36)(26,37)(27,33)(29,35)(34,40)(38,39);(1,4,5,3)(2,6,7,8)(10,16)(11,20)(12,24)(13,25)(15,27)(17,18)(19,34)(21,37)(22,23)(26,39)(28,31)(29,40)(30,32)(33,35)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151502, #151503, #151504, … |
| maximal normal subgroup of | #151502, #151503, #151504, … |
| maximal quotient of | #146143, #146144, #146154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6505);