Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 262144 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,17)(19,20);(2,3)(5,7,8,12)(6,9,10,11);(1,2,4,3)(14,17)(18,20);(1,3,4,2);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=h,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 | #122065, #122079, #122089, … |
| maximal normal subgroup of | #122065, #122079, #122089, … |
| maximal quotient of | #514106, #514224, #514300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27926);