Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | C23 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,20,13,21,22,33,14,19,23,31,24,32,34,39)(10,15,16,26,17,27,28,37,18,25,29,35,30,36,38,40);(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=ef,f^2=g,g^2=h,b^a=be,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #128716, #128728, #134949, … |
| maximal normal subgroup of | #128716, #128728, #134949, … |
| maximal quotient of | #493120, #493126, #493256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26464);