Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| 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 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 123 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)(9,10)(11,21)(12,16)(13,27)(14,31)(15,20)(17,24)(18,29)(19,22)(23,32)(25,34)(26,33)(28,36)(30,35)(37,38)(39,40);(2,5)(6,8)(9,11,22,26,15,24,14,12)(10,16,31,17,20,33,19,21)(13,23,29,38,30,39,28,25)(18,32,27,34,36,40,35,37);(1,3,4,7)(2,6,5,8)(9,12,14,24,15,26,22,11)(10,17,19,16,20,21,31,33)(13,25,28,39,30,38,29,23)(18,34,35,32,36,37,27,40);(9,13)(10,18)(11,23)(12,25)(14,28)(15,30)(16,32)(17,34)(19,35)(20,36)(21,37)(22,29)(24,39)(26,38)(27,31)(33,40)Polycyclic presentation:
2^8|b^2=fh,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6491808, #6491809, #6491817, … |
| maximal normal subgroup of | #6491808, #6491809, #6491817, … |
| maximal quotient of | #350890, #350891, #350896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26050);