Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1670 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)(9,10)(11,13)(12,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,26,30,31)(27,28,33,36)(29,37,39,32)(34,38,40,35);(10,11)(14,15)(17,19,22,23)(18,21,24,20)(26,31)(27,33)(32,37)(34,40);(9,10,13,11)(12,14,16,15)(17,20,22,21)(18,23,24,19)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,14,16,15)(17,21,22,20)(18,23,24,19)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,23)(22,24)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=g,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4816731);