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) = 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 | |
2326 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)(9,10,12,14)(11,13,16,15)(17,18,23,27)(19,25,20,24)(21,22,32,26)(28,31,30,29);(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,27,25)(21,30,32,28)(22,31,26,29);(5,6)(7,8)(17,20,23,19)(18,25,27,24)(21,30,32,28)(22,31,26,29);(1,2,3,4)(9,10)(11,15)(12,14)(13,16)(17,21,23,32)(18,26,27,22)(19,28,20,30)(24,31,25,29);(9,11)(10,13)(12,16)(14,15)(17,22,23,26)(18,21,27,32)(19,29,20,31)(24,28,25,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fgi,e^2=g,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, 2829018);