Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,11)(7,14)(8,9)(12,19)(13,17)(15,16);(1,3,4,2)(5,6)(7,12)(8,17)(9,11)(10,13)(14,18)(15,19)(16,20);(5,7,9,14,10,15,8,16)(6,12,11,18,13,19,17,20);(7,15)(12,19)(14,16)(18,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #367509, #367510, #367548, … |
| maximal normal subgroup of | #367509, #367510, #367548, … |
| maximal quotient of | #203711, #203715, #203719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14170);