Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C23 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,15,26,32,12,17,25,14,9,27,8,23)(7,13,20,28,21,29,18,33,22,30,34,31,19,35,24,36);(2,4)(5,7,10,20,11,21,26,18,12,22,25,34,9,19,8,24)(6,13,16,28,15,29,32,33,17,30,14,31,27,35,23,36);(5,8,9,25,12,26,11,10)(6,14,15,23,17,16,27,32)(7,18,19,20,22,24,21,34)(13,28,29,33,30,31,35,36)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #93930, #135892, #135913, … |
| maximal normal subgroup of | #93930, #135892, #135913, … |
| maximal quotient of | #140336, #140339, #140350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6150);