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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,7)(11,15)(13,16)(17,18,22,21)(19,23,27,26)(20,28,30,24)(25,32,29,31);(1,2,5,4)(3,8,7,6)(9,10,12,14)(11,13,15,16)(18,21)(23,26)(24,28)(31,32);(1,3,5,7)(2,6,4,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,27)(18,23,21,26)(20,29,30,25)(24,32,28,31);(1,4,5,2)(3,8,7,6)(17,20,22,30)(18,24,21,28)(19,25,27,29)(23,31,26,32);(17,21,22,18)(19,26,27,23)(20,28,30,24)(25,32,29,31)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=f,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 | #6679526, #6679531, #6679542, … |
| maximal normal subgroup of | #6679526, #6679531, #6679542, … |
| maximal quotient of | #544438, #545035, #545501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35650);