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 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,15,16)(18,21)(19,20)(27,33)(28,35)(30,36)(31,38);(4,8)(6,7)(10,13)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(2,3)(6,7)(10,13)(14,16)(17,19,22,20)(18,23,21,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,23,21,24)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,18)(19,23,20,24)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=fghi,d^2=gh,e^2=fghi,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, 5380879);