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 | C25 | 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) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(11,15)(13,16)(17,18,22,23)(19,25,27,21)(20,28,31,24)(26,30,29,32);(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,13,15,16)(18,23)(21,25)(24,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,29)(22,27)(23,25)(24,32)(26,31)(28,30);(1,3,4,8)(2,5,6,7)(17,20,22,31)(18,24,23,28)(19,26,27,29)(21,30,25,32);(17,21)(18,19)(20,30)(22,25)(23,27)(24,26)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,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 | #133203, #253309, #253311, … |
| maximal normal subgroup of | #133203, #253309, #253311, … |
| maximal quotient of | #544431, #545028, #545499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34379);