Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1803 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)(5,8)(6,7)(10,13)(11,15)(17,18,22,20)(19,23,21,24)(26,30)(28,32);(1,3)(2,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,20,24)(26,30)(27,31);(1,4,7,5)(2,8,6,3)(9,11,12,15)(10,14,13,16)(17,19,22,21)(18,24,20,23)(25,26)(27,28)(29,30)(31,32);(1,5,7,4)(2,8,6,3)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,23,21,24)(25,27)(26,28)(29,31)(30,32);(1,6,7,2)(3,4,8,5)(9,11)(10,14)(12,15)(13,16)(17,21,22,19)(18,23,20,24)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=fghi,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2213189);