Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
955 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)(17,18,22,21)(19,24,23,20);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,24)(22,23);(2,5)(6,7)(9,10)(11,15)(12,14)(13,16)(17,20,22,24)(18,23,21,19);(1,2)(3,7)(4,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,21,22,18)(19,24,23,20);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)Polycyclic presentation:
2^8|a^2=f,b^2=fgh,c^2=f,d^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 | C2×#36249, #7629906, #7630062, … |
| maximal normal subgroup of | C2×#36249, #7629906, #7630062, … |
| maximal quotient of | #519037, #519039, #519083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29237);