Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
846 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)(4,7)(6,8)(9,10,13,15)(11,12,16,14)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32);(2,3)(6,7)(9,11,13,16)(10,14,15,12)(18,21)(23,25)(24,28)(31,32);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,14)(15,16)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32);(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=g,d^2=g,e^2=fh,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 | #6699934, #6699939, #6699999, … |
| maximal normal subgroup of | #6699934, #6699939, #6699999, … |
| maximal quotient of | #544632, #545165, #545632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35766);