Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
143 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)(5,6,10,15,11,17,25,16)(7,12,9,24,19,14,26,28)(8,20,22,33,23,27,35,34)(13,29,30,36,32,18,31,21);(2,4)(5,7,10,9,11,19,25,26)(6,12,15,24,17,14,16,28)(8,18,22,21,23,29,35,36)(13,27,30,34,32,20,31,33);(5,8,19,29,10,22,26,36,11,23,7,18,25,35,9,21)(6,13,24,34,16,31,12,27,17,32,28,33,15,30,14,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #93933, #93939, #152437, … |
| maximal normal subgroup of | #93933, #93939, #152437, … |
| maximal quotient of | #140341, #140342, #140365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6157);