Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1199 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)(5,6,7,8)(13,14)(15,19)(16,17)(18,20);(9,10,11,12)(14,17)(18,19);(1,2)(3,4)(5,6)(7,8)(13,15,16,20)(14,18,17,19);(13,14)(15,18)(16,17)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=e,b^a=be,c^a=cf,d^b=de100 conjugacy classes — view the full character table.
| maximal subgroup of | #110532, #182648, #183232, … |
| maximal normal subgroup of | #110532, #182648, #183232, … |
| maximal quotient of | C22⋊C4×#95, #153725, #153728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6787);