Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,12,18)(7,10,21,28)(8,19,25,35)(9,17,11,15)(13,16,26,20)(14,30,32,24)(22,27,31,36)(23,33,29,34);(5,7,12,21)(6,13,18,26)(8,22,25,31)(9,16,11,20)(10,15,28,17)(14,23,32,29)(19,30,35,24)(27,34,36,33);(2,4)(5,8,9,23,12,25,11,29)(6,14,15,31,18,32,17,22)(7,19,20,34,21,35,16,33)(10,27,26,24,28,36,13,30)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=d,d^2=h,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=dh,e^b=eg,f^b=fg,g^b=gh,e^c=egh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125723, #125724, #125735, … |
| maximal normal subgroup of | #125723, #125724, #125735, … |
| maximal quotient of | #144894, #144927, #145014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6403);