Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,16)(11,21)(12,13)(14,25)(18,29)(19,30)(20,34)(22,23)(24,36)(26,27)(28,32)(33,39)(35,37)(38,40);(2,5)(6,8)(9,10,13,19,14,20,26,29,15,18,27,34,25,30,12,16)(11,17,21,32,23,33,35,38,24,31,36,40,37,39,22,28);(1,3,4,7)(2,6,5,8)(9,11,12,22,25,37,27,36,15,24,26,35,14,23,13,21)(10,17,16,28,30,39,34,40,18,31,29,38,20,33,19,32);(10,18)(16,29)(17,31)(19,34)(20,30)(28,38)(32,40)(33,39)Polycyclic presentation:
2^8|b^2=f,c^2=e,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #504710, #504712, #504715, … |
| maximal normal subgroup of | #504710, #504712, #504715, … |
| maximal quotient of | #494086, #494908, #494909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26690);