Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,4)(17,18,21,19)(20,23,24,22)(26,30)(28,35)(31,36)(34,39);(1,2,5,6)(3,8,4,7)(9,10,12,14)(11,13,15,16)(17,19,21,18)(20,23,24,22)(26,30)(27,33)(32,37)(34,39);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,14)(15,16)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,7)(4,8)(5,6)(17,20)(18,22)(19,23)(21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,13,14,16)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fi,b^2=ghi,c^2=f,d^2=fg,e^2=fgh,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, 2534054);