Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(6,7)(9,10)(11,15)(12,14)(13,16);(1,2)(3,4)(5,6)(7,8);(2,4)(5,7,8,6)(10,14)(11,12)(15,16);(9,11,13,12)(10,15,16,14)Polycyclic presentation:
2^8|c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #395420, #395421, #395422, … |
| maximal normal subgroup of | #395420, #395421, #395422, … |
| maximal quotient of | #238675, #238678, #238681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16470);