Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,5,4,7)(9,10,13,14)(11,12,15,16)(25,26)(27,33)(28,30)(29,36)(31,35)(32,37)(34,38)(39,40);(1,3,6,4)(2,7,8,5)(10,14)(12,16)(18,21)(22,24)(26,28)(29,37)(31,33)(34,40);(1,3,6,4)(2,5,8,7)(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4,6,3)(2,5,8,7)(9,11)(10,12)(13,15)(14,16)(17,19,20,23)(18,22,21,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,5)(2,4)(3,8)(6,7)(9,12)(10,15)(11,14)(13,16)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2734486);