Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | Aut(G) order 16384 |
| 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 | #3150 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
463 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,28)(22,30)(23,31)(25,34)(27,32)(29,33)(35,37)(36,39)(38,40);(3,7)(5,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,30)(18,21)(19,34)(20,31)(26,39)(27,36)(28,37)(29,38)(32,40)(33,35);(1,2)(3,8)(4,6)(5,7)(9,12,15,29)(10,17,20,32)(11,22,25,38)(13,28,14,27)(16,30,31,40)(18,33,19,26)(21,35,34,39)(23,37,24,36);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,34)(17,26,32,33)(22,36,38,37)(30,39,40,35)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152590, #245315, #245320, … |
| maximal normal subgroup of | #152590, #245315, #245320, … |
| maximal quotient of | #167767, #167802, #167844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12042);