Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,22)(23,24)(26,30)(28,32);(9,11,12,16)(10,13,14,15)(18,19)(22,23)(26,30)(27,31);(10,14)(13,15)(17,18,21,19)(20,23,24,22)(25,26)(27,28)(29,30)(31,32);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,18)(20,23,24,22)(25,27)(26,28)(29,31)(30,32);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,21,24)(18,22,19,23)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fghi,b^2=gh,c^2=fgh,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4768449);