Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(10,11)(12,16)(19,23)(22,24)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,39,37,36);(10,11)(14,15)(18,21)(22,24)(25,27,30,35)(26,28,33,31)(29,34,38,37)(32,36,40,39);(9,10,13,11)(12,14,16,15)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19,20,23)(18,22,21,24)(25,29,30,38)(26,32,33,40)(27,34,35,37)(28,36,31,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fh,d^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4658667);