Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,25)(19,23,27,22)(20,24,29,32)(26,31,28,30);(1,2)(3,7)(4,6)(5,8)(17,19,21,27)(18,22,25,23)(20,28,29,26)(24,31,32,30);(1,2,4,6)(3,5,8,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,28)(21,27)(22,25)(24,30)(26,29)(31,32);(1,2)(3,5)(4,6)(7,8)(17,20,21,29)(18,24,25,32)(19,26,27,28)(22,30,23,31);(1,3)(2,5)(4,8)(6,7)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=g,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46845, #7700737, #7700776, … |
| maximal normal subgroup of | C2×#46845, #7700737, #7700776, … |
| maximal quotient of | #576739, #576742, #576758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39833);