Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,15,16,12)(17,18)(19,20)(21,23)(22,24)(26,30)(28,32);(1,3,5,8)(2,4,6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,21,24,20)(26,30)(27,31);(3,8)(4,7)(10,14)(12,15)(17,20,22,21)(18,19,24,23)(25,26)(27,28)(29,30)(31,32);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,21,22,20)(18,23,24,19)(25,27)(26,28)(29,31)(30,32);(1,4,5,7)(2,3,6,8)(9,12,13,15)(10,16,14,11)(17,18)(19,21)(20,23)(22,24)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fghi,b^2=gh,c^2=ghi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3765210);