Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(19,23)(22,24)(25,26,30,31)(27,28,29,32);(1,2)(3,6)(4,5)(7,8)(17,18,21,20)(19,22,23,24)(26,31)(28,32);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(17,19,21,23)(18,22,20,24)(25,27,30,29)(26,32,31,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28,30,32)(26,27,31,29);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,32,31,28)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836815);