Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| 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 | #2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
287 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,16)(11,20)(12,15)(14,27)(17,31)(19,36)(21,33)(22,39)(23,38)(25,26)(28,29)(30,35)(32,40)(34,37);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,24)(13,16)(14,19)(15,18)(17,23)(20,31)(22,32)(25,36)(26,30)(27,35)(28,40)(29,34)(33,38)(37,39);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,28)(15,29)(16,21)(18,32)(19,37)(20,27)(24,34)(25,38)(26,39)(30,40)(31,36)(33,35);(2,6)(5,7)(9,12,13,15)(10,18,16,24)(11,22,23,29)(14,25,26,27)(17,32,21,34)(19,35,30,36)(20,28,38,39)(31,37,33,40)Polycyclic presentation:
2^8|b^2=eh,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,d^b=dh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33791, #7623352, #7623362, … |
| maximal normal subgroup of | C2×#33791, #7623352, #7623362, … |
| maximal quotient of | #496372, #496488, #496744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26779);