Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(11,15)(13,16)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,32,30,31);(6,8)(9,10,12,14)(11,13,15,16)(18,23)(20,29)(21,25)(26,30);(5,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,30)(22,27)(23,25)(24,31)(26,29)(28,32);(1,2,3,4)(9,10)(11,13)(12,14)(15,16)(17,20,22,29)(18,24,23,28)(19,26,27,30)(21,32,25,31);(9,11)(10,13)(12,15)(14,16)(17,21)(18,19)(20,31)(22,25)(23,27)(24,30)(26,28)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=h,d^2=i,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, 3043360);