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 | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2525 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)(3,6)(4,7)(5,8)(9,10,12,13)(11,15,16,14)(18,21)(19,24)(26,30)(31,32);(1,3,4,8)(2,5,7,6)(10,13)(11,16)(17,18,20,21)(19,22,24,23)(25,26)(27,31)(28,32)(29,30);(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,29,28)(26,31,30,32);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,27)(26,31,30,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=ghi,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, 2958020);