Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(11,17)(14,21)(18,23)(22,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(5,6,7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327357);