Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4)(9,10,13,15)(11,12,14,16)(18,19)(21,29)(23,25)(27,30);(6,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,32,30,31);(5,6)(7,8)(9,11,13,14)(10,12,15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,32,30,31);(1,2)(3,4)(9,10)(11,12)(13,15)(14,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31);(9,12)(10,14)(11,15)(13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2513797);