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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,26,30,31)(27,34,37,32)(28,29,38,33)(35,39,40,36);(1,2,5,4)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23)(26,31)(28,38)(32,34)(35,40);(17,18)(19,22)(20,23)(21,24)(25,27,30,37)(26,32,31,34)(28,35,38,40)(29,36,33,39);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,18)(19,21)(20,23)(22,24)(25,28,30,38)(26,33,31,29)(27,35,37,40)(32,39,34,36);(1,4,5,2)(3,8,7,6)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,24)(21,23)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,35,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=g,d^2=fh,e^2=fghi,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, 2637699);