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 | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
492 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,17)(14,18)(15,19)(16,22)(20,24)(21,23);(1,3,5,4)(2,6,7,8)(9,11,15,21)(10,12,19,23)(13,16,14,20)(17,22,18,24);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,18)(13,22)(14,24)(15,23)(17,21)(19,20);(2,7)(6,8)(9,13,15,14)(10,17,19,18)(11,16,21,20)(12,22,23,24)Polycyclic presentation:
2^8|b^2=eh,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24087, #6452951, #6452953, … |
| maximal normal subgroup of | C2×#24087, #6452951, #6452953, … |
| maximal quotient of | #256508, #256511, #256514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17075);