Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1966 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,19,21,27)(18,22,23,30)(20,25,28,26)(24,29,32,31);(9,12,13,11)(10,14,15,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,3)(2,6,7,8)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^9|b^2=fgi,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354129);