Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3035 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,19)(11,20,12,16)(13,18,15,23)(17,21,24,22);(1,2,3,5,4,6,7,8)(9,11,14,12)(10,16,19,20)(13,22,15,21)(17,18,24,23);(9,12,14,11)(10,17,19,24)(13,22,15,21)(16,18,20,23);(2,6)(5,8)(9,13,14,15)(10,18,19,23)(11,21,12,22)(16,17,20,24)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=h,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19498, #6354576, #6354577, … |
| maximal normal subgroup of | C2×#19498, #6354576, #6354577, … |
| maximal quotient of | #189159, #197894, #197979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12486);