Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2461 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)(11,13)(12,14)(15,16)(17,18,23,27)(19,26,21,24)(20,22,31,25)(28,29,30,32);(6,8)(9,10,12,14)(11,13,15,16)(17,19,23,21)(18,24,27,26)(20,30,31,28)(22,29,25,32);(5,6)(7,8)(11,15)(13,16)(17,20,23,31)(18,25,27,22)(19,28,21,30)(24,29,26,32);(1,2,3,4)(9,10)(11,16)(12,14)(13,15)(17,21,23,19)(18,26,27,24)(20,30,31,28)(22,29,25,32);(9,11)(10,13)(12,15)(14,16)(17,22,23,25)(18,20,27,31)(19,29,21,32)(24,30,26,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3371355);