Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
2047 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)(3,7)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,21)(23,24)(25,26,30,28)(27,32,31,29);(1,2,5,4)(3,8,7,6)(11,16)(13,15)(18,20)(22,23)(25,26,30,28)(27,32,31,29);(2,4)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,27)(26,29)(28,32)(30,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22)(25,28,30,26)(27,32,31,29);(1,4,5,2)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,22)(21,24)(25,29,30,32)(26,31,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038406);