Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19420 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,14,18)(11,15,20,29)(12,22,24,32)(13,25,27,17)(16,26,31,23)(19,33,34,40)(21,37,36,30)(28,38,39,35);(1,2,6,8,4,7,3,5)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,30)(18,29)(22,33)(23,35)(24,34)(25,37)(26,38)(27,36)(31,39)(32,40);(2,7)(5,8)(9,12,14,24)(10,16,18,31)(11,19,20,34)(13,26,27,23)(15,28,29,39)(17,32,25,22)(21,38,36,35)(30,40,37,33);(11,20)(15,29)(19,34)(21,36)(28,39)(30,37)(33,40)(35,38)Polycyclic presentation:
2^8|a^2=h,b^2=eg,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #267902, #267904, #267934, … |
| maximal normal subgroup of | #267902, #267904, #267934, … |
| maximal quotient of | #189145, #197871, #197951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12428);