Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | |
1877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,7,4)(9,10,13,12)(11,15,16,14)(19,22)(20,24)(27,31)(30,32);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,23)(19,20,22,24)(26,29)(30,32);(1,3)(2,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(17,19,21,22)(18,20,23,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27,28,31)(26,30,29,32);(1,4,5,6)(2,7,8,3)(9,12,13,10)(11,15,16,14)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=fhi,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1178572);