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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1650 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)(11,13,15,16)(17,18,23,21)(19,24,22,20)(25,26,30,31)(27,34,33,29)(28,32,36,39)(35,37,40,38);(1,2,5,4)(3,6,7,8)(11,15)(13,16)(17,19,23,22)(18,24,21,20)(26,31)(27,33)(28,36)(37,38);(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,21)(23,24)(25,27,30,33)(26,29,31,34)(28,35,36,40)(32,38,39,37);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,21,23,18)(19,24,22,20)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,14)(15,16)(17,22,23,19)(18,24,21,20)(25,29)(26,33)(27,31)(28,37)(30,34)(32,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=f,d^2=fh,e^2=fhi,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, 2637063);