Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2022 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)(11,13)(12,15)(14,16)(17,18,21,23)(19,24,22,20)(25,26)(27,30)(28,29)(31,32);(2,4)(6,7)(9,10,13,11)(12,15,16,14)(17,19)(18,20)(21,22)(23,24)(25,27,31,29)(26,28,32,30);(3,8)(6,7)(10,11)(14,15)(19,22)(20,24)(25,28,31,30)(26,27,32,29);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14)(17,20,21,24)(18,22,23,19)(25,29,31,27)(26,28,32,30);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,18)(19,24)(20,22)(21,23)(25,30,31,28)(26,27,32,29)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4238603);