Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,22,24,20,26)(19,29,30,32,27,25,31,21);(1,2,5,6)(3,7,4,8)(9,11,12,15)(10,13,14,16)(18,24)(19,27)(26,28)(30,31);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,22,27)(18,25,24,29)(20,31,23,30)(21,26,32,28);(1,4,5,3)(2,7,6,8)(17,20,22,23)(18,26,24,28)(19,30,27,31)(21,29,32,25);(17,21,23,29,22,32,20,25)(18,27,28,31,24,19,26,30)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=dfi,d^c=di,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292903);