Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
817 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 817 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,7)(3,5,8,6)(9,10,13,16)(11,14,12,15);(2,5)(3,8)(6,7)(10,14)(11,12)(15,16);(10,15)(11,12)(14,16);(10,16)(14,15)Polycyclic presentation:
2^8|a^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #245712, #245713, #245724, … |
| maximal normal subgroup of | #245712, #245713, #245724, … |
| maximal quotient of | #165911, #165915, #165920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11735);