Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,7,8)(9,10,12,11)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(9,11,12,10)(14,17)(18,20);(2,4)(10,11);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg88 conjugacy classes — view the full character table.
| maximal subgroup of | #354387, #354388, #354389, … |
| maximal normal subgroup of | #354387, #354388, #354389, … |
| maximal quotient of | #153734, #153736, #153738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6830);