Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1484 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)(6,8)(9,10,14,13)(11,15,16,12)(19,20)(23,24)(25,26,29,30)(27,31,33,34)(28,32,37,38)(35,39,36,40);(9,11,14,16)(10,12,13,15)(18,22)(23,24)(26,30)(27,33)(32,38)(35,36);(10,13)(11,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,20)(18,23,22,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,39)(37,38);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,12,16,15)(17,20,21,19)(18,24,22,23)(25,28,29,37)(26,32,30,38)(27,35,33,36)(31,39,34,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=fh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5258660);