Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26,22,27,21,32)(19,28,29,31,30,24,20,25);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,21)(22,30)(23,29)(25,26)(27,28)(31,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,28)(20,31)(21,26)(22,27)(23,32)(24,30)(25,29);(1,3,5,4)(2,8,6,7)(11,16)(13,15)(17,20,22,29)(18,25,27,31)(19,21,30,23)(24,26,28,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,22,23)(18,26,27,32)(19,29,30,20)(24,31,28,25)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6924863);