Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,8,7,4)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,22)(21,23)(25,26,28,30)(27,31,32,29);(1,3,5,7)(2,4,6,8)(17,19,21,22)(18,20,23,24)(27,32)(29,31);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,23)(21,24)(25,27,28,32)(26,29,30,31);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,23)(22,24);(1,4)(2,7)(3,6)(5,8)(9,11,12,16)(10,13,14,15)(17,20)(18,22)(19,23)(21,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1313511);