Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,19,22,24,21,26)(20,25,29,27,30,32,31,28);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(18,24)(19,26)(25,32)(27,28);(10,14)(13,16)(17,19,23,24,22,26,21,18)(20,28,29,25,30,27,31,32);(1,4,6,5)(2,7,3,8)(17,20,22,30)(18,25,24,32)(19,27,26,28)(21,31,23,29);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,19,24,26)(20,29,30,31)(25,28,32,27)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300525);