Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3867 conjugacy classes of subgroups; only the 1015 normal subgroups are shown.
This group has 3867 conjugacy classes of subgroups — too many to draw. Only its 1015 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16);(10,13)(14,15);(1,2)(3,4)(5,6)(7,8);(9,10)(11,14)(12,13)(15,16);(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^8|a^2=h,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| Frattini subgroup of | #72238708, #72238709 |
| maximal subgroup of | #105849, #106744, #106766, … |
| maximal normal subgroup of | #105849, #106744, #106766, … |
| maximal quotient of | #513545, #513547, #513677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27159);