Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1952 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,23)(19,22,20,24)(25,26,29,30)(27,31,32,28);(2,5)(3,8)(10,13)(11,16)(17,19,21,20)(18,22,23,24)(26,30)(27,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,18)(19,24)(20,22)(21,23)(25,26,29,30)(27,31,32,28);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,22,23,24)(25,27)(26,31)(28,30)(29,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,24)(20,22)(21,23)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=hi,d^2=fh,e^2=gh,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, 4038413);