Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(5,7,8,12)(6,9,10,11);(2,4)(14,17)(18,20);(2,4);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,e^2=g,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #122095, #122196, #370478, … |
| maximal normal subgroup of | #122095, #122196, #370478, … |
| maximal quotient of | #514063, #514221, #514265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27727);