Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(6,7)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,29,33)(27,35,37,31)(28,32,30,34)(36,40,38,39);(1,2,5,3)(4,6,8,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(27,36)(28,30)(31,39)(32,34)(35,40)(37,38);(1,3,5,2)(4,6,8,7)(12,21)(15,24)(18,20)(22,23)(25,27,29,37)(26,31,33,35)(28,36,30,38)(32,39,34,40);(1,4,5,8)(2,6,3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,39,35,40)Polycyclic presentation:
2^8|a^2=ef,b^2=e,c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #123160, #123170, C2×#32851, … |
| maximal normal subgroup of | #123160, #123170, C2×#32851, … |
| maximal quotient of | #391388, #391391, #391426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25839);