Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(11,16)(13,15)(18,21)(19,23)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(9,10,14,12)(11,15,16,13)(18,21)(22,24)(26,30)(31,33)(32,36)(39,40);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,15,12,13)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,c^2=fgh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1028134);