Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1526 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)(9,10,12,14)(11,15,16,13);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(10,14)(13,15)(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(2,5)(6,8)(10,14)(13,15)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fgi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7451236);