Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1616 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,21,23,24)(25,26)(27,29)(28,35)(30,31)(32,37)(33,38)(34,40)(36,39);(9,11,14,13)(10,15,12,16)(19,23)(21,24)(26,31)(27,33)(32,35)(34,39);(1,3,6,5)(2,7,8,4)(9,11,14,13)(10,16,12,15)(17,19)(18,21)(20,23)(22,24)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(1,4,6,7)(2,3,8,5)(9,12,14,10)(11,16,13,15)(25,28,30,37)(26,32,31,35)(27,34,33,39)(29,36,38,40);(1,5,6,3)(2,7,8,4)(9,13,14,11)(10,16,12,15)(17,18)(19,21)(20,22)(23,24)(25,29)(26,27)(28,36)(30,38)(31,33)(32,34)(35,39)(37,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=fg,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4929517);