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 | |
2224 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)(9,10,13,14)(11,17,19,15)(12,20,22,16)(18,23,21,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,40)(36,37);(5,6,7,8)(10,14)(15,17)(16,20)(23,24)(26,30)(27,33)(32,36)(34,35);(5,6)(7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,27,29,33)(26,31,30,38)(28,35,37,34)(32,40,36,39);(1,2)(3,4)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,38)(34,40)(35,39);(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,28,29,37)(26,32,30,36)(27,34,33,35)(31,39,38,40)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=g,e^2=f,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, 3242399);