Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,14)(12,16)(13,15);(2,7)(3,4)(9,11,13,16)(10,14,15,12)(18,22)(20,28)(23,25)(26,32);(1,3,6,4)(2,8,7,5)(9,11)(10,12)(13,16)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,4,6,3)(2,5,7,8)(9,12)(10,11)(13,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5)(2,3)(4,7)(6,8)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132892, #486842, #486970, … |
| maximal normal subgroup of | #132892, #486842, #486970, … |
| maximal quotient of | #585264, #585795, #587382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51974);