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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,14)(11,16)(17,18)(19,24)(20,23)(21,22)(25,26)(27,31)(28,32)(29,30);(9,10)(11,12)(13,14)(15,16)(17,19,22,23)(18,20,21,24)(26,30)(28,31);(10,14)(12,15)(18,21)(19,23)(25,27,29,32)(26,31,30,28);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,15,14,12)(17,20,22,24)(18,23,21,19)(25,28)(26,27)(29,31)(30,32);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,11,14,16)(17,21,22,18)(19,20,23,24)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fgi,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, 4028728);