Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,20,23,22)(25,26)(27,33)(28,36)(29,30)(31,34)(32,38)(35,39)(37,40);(1,2,5,4)(3,6,7,8)(11,16)(13,15)(19,23)(20,22)(26,30)(27,34)(32,36)(35,37);(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,22,24,20)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,22)(18,23,24,19)(25,27)(26,31)(28,37)(29,34)(30,33)(32,40)(35,38)(36,39);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28,29,38)(26,32,30,36)(27,35,34,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=fh,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712087);