Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | Aut(G) order 32768 |
| 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 | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
464 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 464 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,20)(11,21,24,34)(12,17,26,32)(13,19,14,18)(16,28,30,23)(22,35,38,39)(25,33,27,31)(29,36,40,37);(3,7)(5,8)(9,11)(10,16)(12,22)(13,23)(14,28)(15,24)(17,29)(18,21)(19,34)(20,30)(25,36)(26,38)(27,37)(31,39)(32,40)(33,35);(1,2)(3,8)(4,6)(5,7)(9,12,15,26)(10,17,20,32)(11,22,24,38)(13,27,14,25)(16,29,30,40)(18,33,19,31)(21,35,34,39)(23,37,28,36);(9,13,15,14)(10,18,20,19)(11,23,24,28)(12,25,26,27)(16,21,30,34)(17,31,32,33)(22,36,38,37)(29,39,40,35)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152615, #152623, #153154, … |
| maximal normal subgroup of | #152615, #152623, #153154, … |
| maximal quotient of | #168120, #168160, #168208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11707);