Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(11,15)(13,16)(17,18)(19,25)(20,27)(21,29)(22,23)(24,30)(26,28)(31,32);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(18,23)(20,25)(24,29)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6471294, #6471295, #6471354, … |
| maximal normal subgroup of | #6471294, #6471295, #6471354, … |
| maximal quotient of | #553753, #553785, #553980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30382);