Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(1,2,5,8)(3,7,4,6)(9,10,13,18)(11,20,22,16)(12,25,27,35)(14,29,15,19)(17,30,34,28)(21,33,38,40)(23,32,24,37)(26,36,39,31);(1,3,5,4)(2,6,8,7)(9,11,14,23,13,22,15,24)(10,16,29,37,18,20,19,32)(12,26,30,21,27,39,28,38)(17,33,25,31,34,40,35,36);(9,12,14,28,13,27,15,30)(10,17,19,25,18,34,29,35)(11,21,23,26,22,38,24,39)(16,31,32,40,20,36,37,33);(1,4,5,3)(2,7,8,6)(10,18)(16,20)(17,34)(19,29)(25,35)(31,36)(32,37)(33,40)Polycyclic presentation:
2^8|a^2=e,b^2=eg,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139388, #139390, #143289, … |
| maximal normal subgroup of | #139388, #139390, #143289, … |
| maximal quotient of | #325120, #325125, #325132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21689);