Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(10,13)(14,16)(17,18,20,21)(19,23,24,22)(25,26)(27,31)(28,30)(29,32)(33,39)(34,35)(36,38)(37,40);(1,2,4,5)(3,6,7,8)(11,15)(14,16)(18,21)(19,24)(26,28)(31,35)(32,36)(39,40);(2,5)(3,7)(9,10,12,13)(11,14,15,16)(17,19,20,24)(18,22,21,23)(25,27)(26,31)(28,35)(29,37)(30,34)(32,40)(33,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,26)(27,31,34,35)(29,36,38,32)(33,39,37,40);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)(25,29,30,38)(26,32,28,36)(27,33,34,37)(31,39,35,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fghi,c^2=g,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4976374);