Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(6,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(26,30)(27,33)(28,36)(38,40);(10,15)(12,19)(17,23)(20,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,4)(5,6,7,8)(9,11,14,16)(10,12,15,19)(13,21,22,18)(17,24,23,20)(25,26)(27,31)(28,37)(29,30)(32,36)(33,34)(35,40)(38,39);(1,2)(3,4)(9,12)(10,16)(11,15)(13,20)(14,19)(17,21)(18,23)(22,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,21)(19,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4295435);