Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 348 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:
(1,2,4,6)(3,5,7,8)(9,10)(11,23)(12,27)(13,17)(14,21)(15,20)(16,22)(18,29)(19,26)(24,38)(25,33)(28,37)(30,36)(31,35)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,25,16,26,14,13)(10,17,21,19,22,33,20,23)(12,24,31,40,30,39,29,28)(18,32,36,34,35,38,27,37);(3,7)(5,8)(9,12)(10,18)(11,24)(13,28)(14,29)(15,31)(16,30)(17,32)(19,34)(20,27)(21,36)(22,35)(23,37)(25,40)(26,39)(33,38);(9,13,14,26,16,25,15,11)(10,19,20,17,22,23,21,33)(12,28,29,39,30,40,31,24)(18,34,27,32,35,37,36,38)Polycyclic presentation:
2^8|a^2=g,b^2=eh,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139597, #149956, #486672, … |
| maximal normal subgroup of | #139597, #149956, #486672, … |
| maximal quotient of | #314654, #314659, #314663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20661);