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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(17,18,21,19)(20,23,24,22)(26,30)(27,33)(28,35)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,19,21,18)(20,23,24,22)(27,33)(31,36)(34,39)(38,40);(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,7)(4,6)(5,8)(17,20)(18,22)(19,23)(21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,13,14,16)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fghi,c^2=f,d^2=g,e^2=gh,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, 2534064);