Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,11)(14,16);(2,4)(5,6,7,8);(9,10,12,11)(13,14)(15,16);(1,2)(3,4)(5,6)(7,8)(9,11,12,10)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,b^a=be,c^a=cf,d^b=de100 conjugacy classes — view the full character table.
| maximal subgroup of | #219850, #219852, #219864, … |
| maximal normal subgroup of | #219850, #219852, #219864, … |
| maximal quotient of | C22⋊C4×#95, #153727, #153728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6801);