Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,14)(11,16)(17,18,22,23)(19,24,21,20)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,3,6,5)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,21)(18,20,23,24)(28,37)(31,36)(34,40)(38,39);(3,5)(7,8)(10,14)(12,15)(17,20,22,24)(18,19,23,21)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,6,2)(3,8,5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,24)(22,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,16,14,11)(17,21,22,19)(18,20,23,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=fh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4151825);