Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,10,13,23,28)(7,15,17,29,18,11,22,25)(8,19,21,33,20,12,26,27)(16,24,32,35,31,30,34,36);(1,2)(3,4)(5,7,10,18)(6,11,13,15)(8,16,20,31)(9,22,23,17)(12,24,19,30)(14,29,28,25)(21,34,26,32)(27,36,33,35);(2,4)(5,8)(6,12,13,19)(7,16)(9,21)(10,20)(11,24,15,30)(14,27,28,33)(17,32)(18,31)(22,34)(23,26)(25,35,29,36)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94488, #133437, #133438, … |
| maximal normal subgroup of | #94488, #133437, #133438, … |
| maximal quotient of | #118677, #118733, #118735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4156);