Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C23 |
| commutator | C22 | C2×C4×C8 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
587 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 303 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,8)(13,14,15,17,16,18,19,20);(2,4)(9,10,11,12);(5,6,7,8)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,f^2=h,b^a=bd,c^a=ce112 conjugacy classes — view the full character table.
| maximal subgroup of | #92809, #92813, #92814, … |
| maximal normal subgroup of | #92809, #92813, #92814, … |
| maximal quotient of | C4⋊#1273, #108766, #108767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3820);