Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(11,12)(14,15)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2,4,6)(3,5,8,7)(9,10,13,16)(11,14,12,15)(18,21)(22,24)(26,29)(27,33)(32,38)(34,36);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,33)(26,31,29,35)(28,36,37,34)(32,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,20,23)(18,22,21,24)(25,28,30,37)(26,32,29,38)(27,34,33,36)(31,39,35,40);(1,3,4,8)(2,5,6,7)(9,12,13,11)(10,15,16,14)(25,29,30,26)(27,35,33,31)(28,32,37,38)(34,39,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fgi,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3097021);