Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,4,7,6)(11,15)(14,16)(17,18,21,19)(20,22,24,23)(26,30)(27,33)(32,37)(34,39);(1,3,5,7)(2,6,8,4)(10,13)(14,16)(18,19)(22,23)(27,33)(31,36)(34,39)(38,40);(3,7)(4,6)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,22,24,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2)(3,6)(4,7)(5,8)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4,5,6)(2,7,8,3)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=hi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4557657);