Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | Aut(G) order 8192 |
| 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 | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
340 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23);(1,2,3,5,4,6,7,8)(10,14)(15,23)(16,21)(17,24);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17);(2,6)(5,8)Polycyclic presentation:
2^8|b^2=eh,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #267523, #267525, #267527, … |
| maximal normal subgroup of | #267523, #267525, #267527, … |
| maximal quotient of | #189085, #197730, #197735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12162);