Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#546 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,4,3)(6,8)(9,10,14,12)(11,16,13,15);(10,12)(15,16);(5,6)(7,8)(9,11,14,13)(10,15,12,16);(9,12,14,10)(11,16,13,15);(1,3,4,2)(9,13,14,11)(10,16,12,15)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #6383914, #6383916, #6383919, … |
| maximal normal subgroup of | #6383914, #6383916, #6383919, … |
| maximal quotient of | #513624, #513626, #513650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27280);