Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,29,32,31);(9,10,12,14)(11,15,16,13)(18,23)(19,26)(25,28)(27,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3,4,7)(2,5,6,8)(9,11,12,16)(10,13,14,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6548716, #6548717, #6548729, … |
| maximal normal subgroup of | #6548716, #6548717, #6548729, … |
| maximal quotient of | #521615, #522247, #522908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32259);