Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | Aut(G) order 8192 |
| 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 | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 119 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)(9,10,16,21)(11,22,24,23)(12,28,18,14)(13,29,31,19)(15,32,30,20)(17,33,34,25)(26,36,39,37)(27,35,40,38);(2,5)(6,8)(9,11,14,26,15,27,13,25,16,24,28,39,30,40,31,33)(10,17,19,35,20,36,18,22,21,34,29,38,32,37,12,23);(1,3,4,7)(2,6,5,8)(9,12,13,20,30,29,14,10,16,18,31,32,15,19,28,21)(11,23,25,36,40,38,26,17,24,22,33,37,27,35,39,34);(11,24)(17,34)(22,23)(25,33)(26,39)(27,40)(35,38)(36,37)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=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #504711, #504713, #504714, … |
| maximal normal subgroup of | #504711, #504713, #504714, … |
| maximal quotient of | #494086, #494908, #494910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26691);