Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
1627 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)(17,18,23,20,22,26,19,24)(21,25,29,28,30,32,27,31);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,22,23)(18,24,26,20)(21,29,30,27)(25,28,32,31);(10,14)(13,16)(17,20,23,26,22,24,19,18)(21,28,29,32,30,31,27,25);(1,4,6,5)(2,7,3,8)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32);(1,5,6,4)(2,7,3,8)(18,26)(20,24)(21,30)(27,29)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=efi,d^b=dfi,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300526);