Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| 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 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,11)(7,16)(8,17)(9,20)(12,19)(13,22)(14,30)(15,31)(18,27)(21,34)(23,24)(25,29)(26,35)(28,36)(32,33);(5,6,12,26,8,14,18,25,20,29,33,30,34,35,7,11,10,13,16,24,21,28,32,31,9,15,27,36,17,23,19,22);(2,4)(6,13)(11,22)(14,28)(15,29)(23,35)(24,26)(25,31)(30,36)Polycyclic presentation:
2^8|b^2=dfh,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,c^b=ch46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13699, #512938, #512939, … |
| maximal normal subgroup of | C2×#13699, #512938, #512939, … |
| maximal quotient of | #151642, #151651, #151665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6687);