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) = C26 |
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 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21)(26,28)(30,32);(2,6)(4,7)(9,11,13,16)(10,14,12,15)(17,18,20,22)(19,21,24,23)(27,31)(30,32);(17,18)(19,21)(20,22)(23,24)(25,26,29,28)(27,30,31,32);(1,3,5,8)(2,4,6,7)(9,12,13,10)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,32)(29,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,21)(20,24)(22,23)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=gh,d^2=f,e^2=gh,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, 2743732);