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 | #3272 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,17)(11,18,19,24)(12,16,13,21)(15,22,23,20);(2,7)(3,5)(4,8)(11,19)(15,23)(18,24)(20,22);(1,3,6,5)(2,4,8,7)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21);(1,4,6,7)(2,3,8,5)(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=gh,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×#20250, #6359609, #6359654, … |
| maximal normal subgroup of | C2×#20250, #6359609, #6359654, … |
| maximal quotient of | #220908, #220992, #222407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13238);