Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1404 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)(11,13)(12,14)(15,16)(17,18,22,25,21,23,31,29)(19,24,27,20,26,30,32,28);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(18,23)(20,28)(24,30)(25,29);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19,21,26)(18,24,23,30)(20,29,28,25)(22,27,31,32);(2,5)(6,8)(18,23)(19,26)(25,29)(27,32);(17,20,22,30,21,28,31,24)(18,19,25,27,23,26,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cfi,e^b=ei,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285717);