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 | |
1276 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)(17,18,23,27,22,29,21,28)(19,24,20,26,31,25,30,32);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,23)(21,30)(22,31)(25,29)(26,27)(28,32);(9,11)(10,13)(12,16)(14,15)(17,20,22,30)(18,25,29,24)(19,21,31,23)(26,28,32,27);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,21)(22,30)(23,31)(24,28)(25,27)(29,32);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,29,28)(19,20,31,30)(24,32,25,26)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=i,d^2=f,e^2=fi,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, 6923336);