Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15412 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3683 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)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,19)(21,29)(23,25)(27,32);(9,11)(10,13)(12,16)(14,15)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30);(2,3)(10,12)(11,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,4,2)(5,6)(7,8)(9,12,14,10)(11,16,15,13)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(9,13,14,16)(10,15,12,11)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4852089);