Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | Aut(G) order 65536 |
| 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 | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 289 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,16,22)(11,18,23,35)(12,17,26,32)(13,20,29,38)(14,19,30,37)(15,31,28,21)(24,36,39,34)(25,40,27,33);(2,5)(6,8)(9,11,14,24,15,25,29,26,16,23,30,39,28,27,13,12)(10,17,20,33,21,34,37,35,22,32,38,40,31,36,19,18);(1,3,4,7)(2,6,5,8)(9,12,13,27,28,39,30,23,16,26,29,25,15,24,14,11)(10,18,19,36,31,40,38,32,22,35,37,34,21,33,20,17);(11,23)(12,26)(17,32)(18,35)(24,39)(25,27)(33,40)(34,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=e,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137519, #137573, #137576, … |
| maximal normal subgroup of | #137519, #137573, #137576, … |
| maximal quotient of | #494087, #494916, #495124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26693);