Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | |
2035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,12)(13,16)(14,15)(17,18,21,24)(19,22,23,20)(25,26)(27,31)(28,29)(30,32);(1,3,6,8)(2,7,4,5)(9,10,13,16)(11,12,15,14)(17,18)(19,22)(20,23)(21,24)(26,29)(30,31);(1,2,6,4)(3,7,8,5)(9,11,13,15)(10,14,16,12)(17,19,21,23)(18,22,24,20)(25,26)(27,30)(28,29)(31,32);(1,4,6,2)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,20,21,22)(18,23,24,19);(1,5)(2,3)(4,8)(6,7)(9,12,13,14)(10,15,16,11)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fghi,c^2=fgh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 761356);