Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,5)(6,7)(9,10)(11,15)(12,16)(13,14)(25,26,29,30)(27,33,34,31)(28,32,36,37)(35,39,40,38);(2,7)(5,8)(10,13)(11,16)(17,18,21,24)(19,22,20,23)(26,30)(27,34)(28,36)(38,39);(1,3,6,4)(2,8,7,5)(9,11,14,16)(10,15,13,12)(17,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,6,3)(2,5,7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,20)(18,22,24,23)(25,27,29,34)(26,31,30,33)(28,35,36,40)(32,38,37,39);(1,5)(2,3)(4,7)(6,8)(9,13,14,10)(11,12,16,15)(17,20,21,19)(18,23,24,22)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=fg,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1953890);