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 | |
1421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(18,23)(19,27)(22,25)(24,28)(29,31)(30,32);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,24,29,32)(20,22,26,25)(27,30,31,28);(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,24,23,32)(20,30,26,28)(22,31,25,27);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,21,26)(18,25,23,22)(19,28,29,30)(24,31,32,27);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,23)(20,26)(24,32)(28,30)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 6909664);