Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1563 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)(17,18,22,27)(19,25,24,23)(20,29,31,30)(21,26,32,28);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,24)(18,23,27,25)(20,26,31,28)(21,30,32,29);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,25,22,31,32,23)(18,24,28,29,27,19,26,30);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,19)(18,25)(20,26)(21,30)(22,24)(23,27)(28,31)(29,32);(10,14)(13,15)(17,21,22,32)(18,26,27,28)(19,29,24,30)(20,25,31,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=i,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=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6931330);