Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,22,24,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,39)(38,40);(1,2,5,4)(3,6,7,8)(11,16)(13,15)(17,19,21,24)(18,22,20,23)(26,31)(27,34)(33,39)(35,40);(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27,30,34)(26,32,31,29)(28,35,37,40)(33,36,39,38);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22)(25,29,30,32)(26,34,31,27)(28,36,37,38)(33,35,39,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=f,d^2=g,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2630704);