Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1492 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,27,21,26)(19,28,25,31,29,20,30,32);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,31)(21,23)(26,27)(28,32);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,20,27,28)(21,30,23,25)(24,32,26,31);(3,8)(5,7)(11,16)(13,15)(17,20,23,32,22,28,21,31)(18,25,24,29,27,30,26,19);(10,14)(13,15)(17,21,22,23)(18,26,27,24)(19,25,29,30)(20,32,28,31)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6990116);