Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,8)(9,10)(11,16)(12,15)(13,14);(5,6,7,8)(9,11)(10,12)(13,15)(14,16);(1,2)(3,4)(10,14)(11,15);(9,10)(11,12)(13,14)(15,16);(9,12)(10,15)(11,14)(13,16)Polycyclic presentation:
2^8|b^2=fh,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #6573841, #6573845, #6573857, … |
| maximal normal subgroup of | #6573841, #6573845, #6573857, … |
| maximal quotient of | D4×#386, #518973, #518980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29202);