Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×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 | |
300 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 103 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,12,15)(13,16,21,24)(19,23,20,22)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,12)(10,15,17,18)(13,20,21,19)(16,23,24,22)(25,27,29,35)(26,30,31,38)(28,33,36,34)(32,37,40,39);(2,6)(5,8)(9,12,14,11)(10,15,17,18)(13,19,21,20)(16,23,24,22)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(3,7)(5,8)(9,13,14,21)(10,16,17,24)(11,19,12,20)(15,22,18,23)(26,31)(30,38)(32,40)(37,39)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=e,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #143001, #143004, #143005, … |
| maximal normal subgroup of | #143001, #143004, #143005, … |
| maximal quotient of | #391336, #391349, #391399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25782);