Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,8,7,4)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10,13,14)(11,15,12,16)(18,19)(21,29)(23,25)(27,32);(1,4,6,8)(2,7,5,3)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(1,2,6,5)(3,4,7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,5,6,2)(3,8,7,4)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6442153, #6442159, #6442170, … |
| maximal normal subgroup of | #6442153, #6442159, #6442170, … |
| maximal quotient of | #515377, #515415, #515494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28641);