Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,8,7)(11,15)(14,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32);(2,6)(4,7)(9,10,13,12)(11,14,15,16)(18,23)(19,26)(25,28)(27,32);(2,6)(3,8)(9,11,13,15)(10,14,12,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(1,3,5,8)(2,7,6,4)(9,12,13,10)(11,16,15,14)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,4,5,7)(2,8,6,3)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41579, #7640048, #7640233, … |
| maximal normal subgroup of | C2×#41579, #7640048, #7640233, … |
| maximal quotient of | #544721, #545209, #545718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34567);