Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C2×#392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(9,10,12,11)(14,15)(18,19);(6,8)(13,14,17,15)(16,19,20,18);(1,2)(3,4)(5,6)(7,8)(9,11,12,10)(13,15,17,14)(16,19,20,18);(1,2)(3,4)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=eh,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#14520, #520979, #521035, … |
| maximal normal subgroup of | C2×#14520, #520979, #521035, … |
| maximal quotient of | #153970, #153979, #153988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7508);