Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1930 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)(3,7)(10,11)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(17,18,21,24)(19,20,23,22)(29,37)(31,38)(34,40)(36,39);(10,11)(12,16)(19,23)(20,22)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,24,20)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,22)(18,23,24,19)(25,29)(26,31)(27,34)(28,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6040025);