Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,10,8)(7,12,11,9)(13,14)(15,19)(16,18)(17,20);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,17)(14,19,16,20);(5,6,10,8)(7,9,11,12)(13,16,18,14)(15,19,17,20);(1,2)(3,4)(5,8,10,6)(7,12,11,9)(13,17,18,15)(14,19,16,20);(5,9,10,12)(6,11,8,7)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6996829, #6996830, #6996853, … |
| maximal normal subgroup of | #6996829, #6996830, #6996853, … |
| maximal quotient of | #531701, #531756, #531768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52136);