Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
519 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,14)(13,17)(15,22)(18,23)(19,24)(20,21);(2,5)(6,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22);(9,12,14,10)(11,18,20,24)(13,22,15,17)(16,23,21,19);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,17,12,22)(11,19,20,23)(16,24,21,18)Polycyclic presentation:
2^8|c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24149, #6448660, #6448662, … |
| maximal normal subgroup of | C2×#24149, #6448660, #6448662, … |
| maximal quotient of | #257976, #257982, #257988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17137);