Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 532 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)(3,7)(4,5)(6,8)(9,10,13,17)(11,19,21,15)(12,16,24,32)(14,27,26,18)(20,29,36,35)(22,30,28,34)(23,37,38,31)(25,40,39,33);(2,5)(6,7)(9,11)(10,15)(12,23)(13,21)(14,28)(16,31)(17,19)(18,34)(20,25)(22,26)(24,38)(27,30)(29,33)(32,37)(35,40)(36,39);(9,12)(10,16)(11,20,21,36)(13,24)(14,25)(15,29,19,35)(17,32)(18,33)(22,23,28,38)(26,39)(27,40)(30,31,34,37);(1,3,4,8)(2,6,5,7)(10,17)(12,24)(15,19)(18,27)(20,36)(23,38)(25,39)(30,34)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268306, #268311, #268330, … |
| maximal normal subgroup of | #268306, #268311, #268330, … |
| maximal quotient of | #293334, #293337, #293342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18584);