Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 155 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:
(2,4)(5,6)(7,8)(9,10,14,18)(11,19,23,33)(12,16,24,30)(13,26,20,17)(15,27,29,22)(21,32,35,40)(25,37,34,31)(28,38,39,36);(9,11,20,27,14,23,13,22)(10,15,26,33,18,29,17,19)(12,21,34,38,24,35,25,36)(16,28,37,40,30,39,31,32);(1,2,3,4)(9,12)(10,16)(11,21)(13,25)(14,24)(15,28)(17,31)(18,30)(19,32)(20,34)(22,36)(23,35)(26,37)(27,38)(29,39)(33,40);(6,8)(12,24)(16,30)(21,35)(25,34)(28,39)(31,37)(32,40)(36,38)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=f,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125159, #125171, #125444, … |
| maximal normal subgroup of | #125159, #125171, #125444, … |
| maximal quotient of | #351718, #351719, #351728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26168);