Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
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 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,30)(21,25)(22,27)(23,24)(26,29)(31,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,23)(18,24,27,28)(20,31,21,29)(25,32,30,26);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,22,21)(18,25,27,30)(19,29,23,31)(24,26,28,32);(17,21,22,20)(18,26,27,32)(19,29,23,31)(24,25,28,30)Polycyclic presentation:
2^8|a^2=eh,b^2=f,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,c^b=ch,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6472294, #6472304, #6472311, … |
| maximal normal subgroup of | #6472294, #6472304, #6472311, … |
| maximal quotient of | #392296, #392771, #392773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26065);