Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,14,10,11,8,13)(7,12,15,20,16,18,17,19);(6,11)(7,15)(8,9)(12,19)(16,17)(18,20);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,19)(14,20)Polycyclic presentation:
2^8|a^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11014, #357567, #358178, … |
| maximal normal subgroup of | C2×#11014, #357567, #358178, … |
| maximal quotient of | #111393, #111394, #111402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4002);