Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1802 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)(9,10,13,16)(11,12,15,14)(17,18,21,19)(20,22,24,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(18,19)(20,24)(28,36)(31,38)(34,40)(37,39);(3,8)(5,7)(11,15)(12,14)(17,19,21,18)(20,22,24,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,4,8)(2,5,6,7)(9,12,13,14)(10,15,16,11)(17,20,21,24)(18,22,19,23)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=fg,c^2=fgi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5398235);