Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
921 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,13,15)(11,12,16,14)(17,18)(19,23)(20,24)(21,22);(1,2)(3,8)(4,5)(6,7)(9,10,13,15)(11,12,16,14)(18,22)(20,23);(2,5)(6,8)(9,11,13,16)(10,14,15,12)(17,19,21,24)(18,23,22,20);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,21,23)(18,24,22,19);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#43176, #7667409, #7667451, … |
| maximal normal subgroup of | C2×#43176, #7667409, #7667451, … |
| maximal quotient of | #535454, #535511, #536057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36164);