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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(11,16)(13,15)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(9,10,14,12)(11,15,16,13)(17,18,20,22)(19,21,24,23)(26,30)(31,38)(32,35)(39,40);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,15,12,13)(17,18)(19,21)(20,22)(23,24)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=fgh,d^2=fgh,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, 1027655);