Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,7,5,8)(10,14)(12,16)(20,24)(22,23)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(1,3,6,5)(2,7,4,8)(10,14)(11,15)(17,18,21,19)(20,22,24,23)(26,28)(27,33)(32,36)(35,37);(1,3,6,5)(2,8,4,7)(9,10)(11,12)(13,14)(15,16)(17,19,21,18)(20,23,24,22)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,4,6,2)(3,7,5,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,26)(27,34,33,31)(29,32,38,36)(35,39,37,40);(1,5,6,3)(2,7,4,8)(9,12)(10,11)(13,16)(14,15)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=fghi,c^2=fghi,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3370959);