Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| 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 | #3399 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
322 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)(3,6)(4,7)(5,8)(9,10,14,12)(11,17,18,24)(13,21,23,16)(15,22,20,19);(1,3,5,4)(2,6,8,7)(9,11,14,18)(10,15,12,20)(13,22,23,19)(16,24,21,17);(2,7)(3,4)(6,8)(11,18)(15,20)(17,24)(19,22);(2,8)(6,7)(9,12,14,10)(11,17,18,24)(13,16,23,21)(15,22,20,19)Polycyclic presentation:
2^8|a^2=g,b^2=gh,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,c^b=cgh,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20313, #6361254, #6361267, … |
| maximal normal subgroup of | C2×#20313, #6361254, #6361267, … |
| maximal quotient of | #220923, #220933, #220941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13301);