Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
1780 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)(11,15)(13,16)(19,23)(21,24)(27,31)(30,32);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(25,26,29,28)(27,30,31,32);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,21,23,24)(25,27)(26,30)(28,32)(29,31);(17,19)(18,21)(20,23)(22,24)(25,28,29,26)(27,32,31,30);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,15)(14,16)Polycyclic presentation:
2^9|b^2=gh,c^2=fi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4323699);