Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,22,23,20)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,39,40,38);(2,5)(6,8)(17,19,21,23)(18,22,24,20)(26,30)(28,37)(31,33)(34,40);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,21,22)(18,23,24,19)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(17,18)(19,22)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1022496);