Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40);(9,11,13,16)(10,14,15,12)(17,19,22,20)(18,23,21,24)(26,30)(27,33)(32,35)(34,36);(9,12)(10,11)(13,14)(15,16)(17,18,22,21)(19,24,20,23)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,22,19)(18,23,21,24)(25,27,29,33)(26,31,30,38)(28,36,37,34)(32,40,35,39);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,16)(14,15)(17,21,22,18)(19,23,20,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fh,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028727);