Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,11)(8,12)(13,14)(15,18)(16,20)(17,19)(21,22,26,27)(23,29,30,24)(25,33,34,28)(31,32,36,35);(5,6,10,9)(7,8,11,12)(13,15,17,20)(14,18,19,16)(22,27)(23,30)(25,34)(32,35);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23,26,30)(22,24,27,29)(25,31,34,36)(28,32,33,35);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(21,24)(22,23)(25,32)(26,29)(27,30)(28,31)(33,36)(34,35);(5,9,10,6)(7,12,11,8)(13,16)(14,15)(17,18)(19,20)(21,25,26,34)(22,28,27,33)(23,31,30,36)(24,32,29,35)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4226037);