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 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(10,13)(14,16)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30);(3,5)(7,8)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30);(1,3,6,5)(2,7,4,8)(9,10,12,13)(11,14,15,16)(17,20,23,19)(18,24,26,22)(21,27,31,28)(25,30,29,32);(1,4,6,2)(3,7,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29);(1,5,6,3)(2,7,4,8)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,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 | #6702484, #6702489, #6702549, … |
| maximal normal subgroup of | #6702484, #6702489, #6702549, … |
| maximal quotient of | #544655, #545178, #545614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35741);