Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #383 |
| prime radical | C23 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,3,5,7,8)(9,10)(11,19,22,23)(12,15)(13,17)(14,18)(16,21,24,20);(10,15,18,17)(16,23,24,19);(2,5)(6,8)(9,11,12,20,14,22,13,21)(10,16,17,19,18,24,15,23)Polycyclic presentation:
2^8|a^2=f,b^2=deh,c^2=d,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,f^c=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93354, #93618, #93619, … |
| maximal normal subgroup of | #93354, #93618, #93619, … |
| maximal quotient of | #99292, #108408, #108490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1789);