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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
815 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)(11,12)(13,14)(15,16)(17,18,22,23)(19,27,29,24)(20,30,31,25)(21,26,32,28);(1,2)(3,7)(4,5)(6,8)(9,11,13,15)(10,12,14,16)(18,23)(21,32)(24,27)(25,30);(2,5)(6,7)(10,14)(11,15)(17,19,22,29)(18,24,23,27)(20,26,31,28)(21,25,32,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20,22,31)(18,25,23,30)(19,28,29,26)(21,27,32,24);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,21)(18,26)(19,25)(20,27)(22,32)(23,28)(24,31)(29,30)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,d^2=f,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 | #133257, #253292, #253294, … |
| maximal normal subgroup of | #133257, #253292, #253294, … |
| maximal quotient of | #544444, #545029, #545519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33535);