Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| 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 | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,14,19)(11,20,22,12)(13,18,23,17)(15,16,24,21);(2,5)(6,7)(9,10,13,17,14,19,23,18)(11,20,21,24,22,12,16,15);(9,11,13,21,14,22,23,16)(10,15,17,20,19,24,18,12);(1,3,4,8)(2,6,5,7)(9,12,13,15,14,20,23,24)(10,16,17,11,19,21,18,22)Polycyclic presentation:
2^8|a^2=ef,b^2=f,c^2=f,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26503, #6444725, #6444726, … |
| maximal normal subgroup of | C2×#26503, #6444725, #6444726, … |
| maximal quotient of | #308985, #308986, #308989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19491);