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 | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,5,8)(3,7,4,6)(9,10,13,18)(11,20,22,16)(12,24,26,36)(14,28,15,19)(17,30,35,27)(21,38,39,34)(23,32,29,33)(25,31,40,37);(1,3,5,4)(2,6,8,7)(9,11,13,22)(10,16,18,20)(12,25,26,40)(14,23,15,29)(17,34,35,38)(19,32,28,33)(21,30,39,27)(24,37,36,31);(1,3,5,4)(2,7,8,6)(9,12,15,30,13,26,14,27)(10,17,28,36,18,35,19,24)(11,21,29,25,22,39,23,40)(16,31,33,38,20,37,32,34);(1,4,5,3)(2,7,8,6)(10,18)(12,26)(16,20)(19,28)(21,39)(25,40)(27,30)(32,33)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=ef,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^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268308, #268312, #268332, … |
| maximal normal subgroup of | #268308, #268312, #268332, … |
| maximal quotient of | #304369, #304371, #304377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18915);