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 | |
769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,13,16,15)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31);(2,6)(4,7)(9,10,12,14)(11,15,16,13)(18,21)(23,25)(24,28)(30,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31);(1,3,5,8)(2,7,6,4)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(1,4,5,7)(2,8,6,3)(9,11,12,16)(10,13,14,15)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=fgh,e^2=fh,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 | #6698367, #6698370, #6698430, … |
| maximal normal subgroup of | #6698367, #6698370, #6698430, … |
| maximal quotient of | #522052, #522556, #523193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33118);