Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12525 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,14,18)(11,19,23,35)(12,24,26,39)(13,27,25,17)(15,28,30,21)(16,31,32,20)(22,38,37,34)(29,36,33,40);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,11,20,36,13,22,26,28,14,23,31,40,25,37,12,21)(10,15,24,34,17,29,32,35,18,30,39,38,27,33,16,19)Polycyclic presentation:
2^8|a^2=h,b^2=d,c^2=eg,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,c^b=cf46 conjugacy classes — view the full character table.
| maximal subgroup of | #123343, #123344, #123345, … |
| maximal normal subgroup of | #123343, #123344, #123345, … |
| maximal quotient of | #135786, #141712, #141726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5516);