Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,4,7,5)(10,14)(12,16)(18,21)(19,23)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(1,3,6,7)(2,4,8,5)(18,21)(22,24)(26,31)(27,33)(29,37)(36,39);(1,4,6,5)(2,3,8,7)(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,5,6,4)(2,7,8,3)(9,11,13,15)(10,12,14,16)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fgh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1040923);