Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(9,10)(11,13)(12,14)(15,16)(18,19)(20,24)(27,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,11,14,16)(10,15,12,13)(18,19)(22,23);(2,6)(3,8)(10,12)(11,16)(17,18,21,19)(20,22,24,23)(25,26,29,28)(27,30,31,32);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,15,16,13)(17,19,21,18)(20,23,24,22)(25,27,29,31)(26,30,28,32);(1,4,5,7)(2,3,6,8)(9,13,14,15)(10,11,12,16)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|b^2=fgh,c^2=fghi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3238831);