Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C24 · Out(G) = C211 |
| 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 | #2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(6,9)(10,12)(13,15,17,16)(14,18,20,19);(2,4)(13,15,17,16)(14,19,20,18);(2,4)(13,16,17,15)(14,19,20,18);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6384855, #6384857, #6384859, … |
| maximal normal subgroup of | #6384855, #6384857, #6384859, … |
| maximal quotient of | #514090, #514196, #514276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27723);