Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(9,10,17,24)(11,25,30,40)(12,31,28,38)(13,21,32,22)(14,20,15,35)(16,36,26,23)(18,34,39,27)(19,33,37,29);(5,6)(7,8)(9,11,17,30)(10,18,24,39)(12,32,28,13)(14,27,15,34)(16,33,26,29)(19,20,37,35)(21,25,22,40)(23,31,36,38);(2,4)(6,8)(9,12,13,11,26,27,15,33,17,28,32,30,16,34,14,29)(10,19,20,39,36,40,22,38,24,37,35,18,23,25,21,31)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=d,d^2=gh,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,d^b=dg,e^b=egh,f^b=fgh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93849, #93851, #151285, … |
| maximal normal subgroup of | #93849, #93851, #151285, … |
| maximal quotient of | #145370, #145386, #145459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6438);