Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #550 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,6,4,7)(10,15)(11,13)(16,18)(17,23)(19,22)(20,24);(1,3,5,4)(2,6,8,7)(9,10,14,18)(11,15,13,16)(12,20,21,23)(17,22,24,19);(2,7)(3,4)(6,8)(9,11,14,13)(10,16,18,15)(12,19,21,22)(17,20,24,23);(9,12,14,21)(10,17,18,24)(11,19,13,22)(15,23,16,20)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cgh,d^b=de,e^b=eg,f^c=fh,e^d=eg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20204, #6359558, #6359655, … |
| maximal normal subgroup of | C2×#20204, #6359558, #6359655, … |
| maximal quotient of | #220907, #221005, #222429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13192);