Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
2082 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)(11,13)(12,15)(14,16)(17,18,22,21)(19,23,20,24)(25,26)(27,30)(28,32)(29,31);(1,2,4,6)(3,5,8,7)(10,11)(12,16)(18,21)(23,24)(25,26,29,31)(27,32,28,30);(9,10,13,11)(12,14,16,15)(17,19,22,20)(18,23,21,24)(25,27,29,28)(26,30,31,32);(1,3,4,8)(2,5,6,7)(9,11,13,10)(12,14,16,15)(17,20,22,19)(18,23,21,24)(25,28,29,27)(26,30,31,32);(1,2)(3,7)(4,6)(5,8)(9,12,13,16)(10,14,11,15)(17,21,22,18)(19,24,20,23)(25,26)(27,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,c^2=i,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2973715);