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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1945 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(19,24)(21,23)(25,26,28,30)(27,31,32,29);(1,2,4,6)(3,5,7,8)(11,16)(13,15)(17,18,22,20)(19,21,24,23)(25,26,28,30)(27,29,32,31);(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,24)(18,23,20,21)(25,27)(26,29)(28,32)(30,31);(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,22,18)(19,21,24,23);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,21)(18,19)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=gh,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, 3964459);