Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
734 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)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,3,6,4)(2,7,8,5)(18,22)(23,25)(24,29)(30,31);(2,8)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4,6,3)(2,5,8,7)(9,11,12,15)(10,14,13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5)(2,3)(4,8)(6,7)(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=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6486195, #6486196, #6486209, … |
| maximal normal subgroup of | #6486195, #6486196, #6486209, … |
| maximal quotient of | #556436, #556437, #556450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30559);