Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,13,12)(11,15,16,14)(18,19)(20,24)(25,26,29,32)(27,30,28,31);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(18,19)(22,23);(3,7)(6,8)(11,16)(14,15)(17,18,21,19)(20,22,24,23)(25,27,29,28)(26,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,14,16,15)(17,19,21,18)(20,23,24,22)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,24)(18,22,19,23)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=hi,c^2=fghi,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456237);