Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2140 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)(4,7)(6,8)(9,10,14,12)(11,16,13,15)(18,19)(20,24)(25,26,29,31)(27,30,28,32);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(20,24)(22,23)(25,27,29,28)(26,30,31,32);(4,8)(6,7)(11,13)(15,16)(17,18,21,19)(20,22,24,23)(25,26)(27,32)(28,30)(29,31);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,15,13,16)(17,19,21,18)(20,23,24,22)(25,28,29,27)(26,30,31,32);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,16,12,15)(17,20)(18,22)(19,23)(21,24)(25,26)(27,32)(28,30)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fhi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038145);