Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2098 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)(11,12)(13,15)(14,16)(17,18,21,23)(19,22,24,20)(25,26)(27,29)(28,30)(31,32);(2,3)(6,7)(9,11,14,13)(10,15,16,12)(17,19,21,24)(18,20,23,22)(26,28)(27,31);(17,19,21,24)(18,20,23,22)(25,27)(26,29)(28,32)(30,31);(1,3,5,2)(4,7,8,6)(9,12,14,15)(10,11,16,13)(17,19)(18,22)(20,23)(21,24)(25,28,30,26)(27,32,31,29);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,16,12)(17,20,21,22)(18,19,23,24)(25,29,30,32)(26,31,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836502);