Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,12)(11,14,15,16)(17,18,22,20)(19,23,21,24)(26,30)(28,36)(31,33)(34,40);(2,4)(6,7)(11,15)(14,16)(17,19,22,21)(18,23,20,24)(28,36)(32,37)(34,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,13,15)(10,14,12,16)(17,20,22,18)(19,23,21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,20,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,2)(3,6,8,7)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4078711);