Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3251 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,21)(13,23)(14,17)(15,20)(16,22)(18,24);(1,2)(3,6)(4,8)(5,7)(9,11,14,16)(10,15,17,12)(13,21,24,20)(18,22,23,19);(1,3,5,4)(2,6,7,8)(9,12,14,15)(10,16,17,11)(13,19,24,22)(18,20,23,21);(2,7)(6,8)(10,17)(11,16)(18,23)(20,21)Polycyclic presentation:
2^8|b^2=h,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18824, #6348769, #6348806, … |
| maximal normal subgroup of | C2×#18824, #6348769, #6348806, … |
| maximal quotient of | #168923, #168979, #169023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11812);