Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
2184 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)(5,6)(7,8)(9,10,14,15)(11,16,18,13)(12,20,23,17)(19,22,21,24)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(10,15)(13,16)(17,20)(22,24)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(6,7)(9,11,14,18)(10,16,15,13)(12,21,23,19)(17,22,20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(1,2)(3,4)(5,7,8,6)(9,12,14,23)(10,17,15,20)(11,19,18,21)(13,22,16,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(9,13,14,16)(10,18,15,11)(12,22,23,24)(17,21,20,19)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4843518);