Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1518 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,28,22,24,21,27)(19,29,30,20,26,25,32,31);(2,5)(6,8)(18,24)(20,31)(25,29)(27,28);(9,11,13,12)(10,14,15,16)(17,19,22,26)(18,25,24,29)(20,28,31,27)(21,32,23,30);(9,12,13,11)(10,14,15,16)(17,20,21,29,22,31,23,25)(18,26,27,30,24,19,28,32);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,24,28)(19,30,26,32)(20,25,31,29)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^b=di,e^c=ei,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281281);