Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24)(27,32)(29,31);(2,5)(6,8)(10,12)(11,15)(17,19,21,24)(18,22,20,23)(25,26,30,28)(27,29,32,31);(3,7)(6,8)(10,12)(14,16)(18,20)(22,23)(25,27,30,32)(26,31,28,29);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)(25,28,30,26)(27,29,32,31);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,14,15,16)(17,20,21,18)(19,22,24,23)(25,29)(26,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=h,d^2=h,e^2=f,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, 3964460);