Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
2338 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)(9,10,13,14)(11,17,18,15)(12,20,22,16)(19,23,21,24)(25,26,31,34)(27,32,28,37)(29,40,33,30)(35,36,38,39);(6,8)(10,14)(11,18)(16,20)(19,21)(25,27,31,28)(26,32,34,37)(29,38,33,35)(30,36,40,39);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28,31,27)(26,32,34,37)(29,38,33,35)(30,39,40,36);(1,2)(3,4)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,29,31,33)(26,30,34,40)(27,35,28,38)(32,39,37,36);(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,30)(26,33)(27,36)(28,39)(29,34)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2513855);