Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,11)(12,13,16,15)(17,18,19,21,20,22,23,24);(2,5)(6,8)(9,11,13,16,14,10,15,12);(1,3,4,7)(2,6,5,8)(9,12,13,11,14,16,15,10)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93440, #93617, #93650, … |
| maximal normal subgroup of | #93440, #93617, #93650, … |
| maximal quotient of | #99282, #108514, #108783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1774);