Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 32768 |
| 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 | C23 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(1,2,5,8)(3,7,4,6)(9,10,13,17)(11,20,22,15)(12,24,26,35)(14,19,29,18)(16,28,34,27)(21,37,38,33)(23,31,40,32)(25,30,39,36);(1,3,5,4)(2,6,8,7)(9,11,14,23,13,22,29,40)(10,15,19,32,17,20,18,31)(12,25,28,38,26,39,27,21)(16,33,24,36,34,37,35,30);(9,12,14,27,13,26,29,28)(10,16,18,24,17,34,19,35)(11,21,23,39,22,38,40,25)(15,30,31,33,20,36,32,37);(1,4,5,3)(2,7,8,6)(10,17)(15,20)(16,34)(18,19)(24,35)(30,36)(31,32)(33,37)Polycyclic presentation:
2^8|a^2=e,b^2=egh,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #143889, #143890, #143893, … |
| maximal normal subgroup of | #143889, #143890, #143893, … |
| maximal quotient of | #325121, #325126, #325131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21690);