Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10)(11,14)(12,17)(13,18)(15,20)(16,22)(19,23)(21,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(6,8)(10,11)(15,17)(16,18)(23,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(5,6,7,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,2)(3,4)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3244628);