Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(18,23)(19,24);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,18)(19,20)(21,24)(22,23);(2,5)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,24)(18,20,23,21);(1,2)(3,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,21)(18,19,23,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,21,22,20)(18,24,23,19)Polycyclic presentation:
2^8|b^2=fgh,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #6580280, #6580285, #6580345, … |
| maximal normal subgroup of | #6580280, #6580285, #6580345, … |
| maximal quotient of | #519006, #519032, #519074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29244);