Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19304 |
| 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 | #3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
327 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)(3,6)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,2,6,8,4,7,3,5);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(2,7)(5,8)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|b^2=eg,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19310, #6353968, #6353991, … |
| maximal normal subgroup of | C2×#19310, #6353968, #6353991, … |
| maximal quotient of | #189123, #197808, #197811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12298);