Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 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)(11,12)(14,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(9,10)(11,16)(12,14)(13,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(2,5)(6,8)(9,11,13,12)(10,14,15,16);(10,15)(14,16)(18,23)(22,31)(24,28)(25,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)Polycyclic presentation:
2^9|b^2=i,c^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6387420);