Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1485 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)(17,18)(19,29)(20,25)(21,28)(22,27)(23,26)(24,30)(31,32);(1,3,5,4)(2,6,8,7)(9,10,14,12)(11,16,13,15)(17,19)(18,24)(20,23)(21,31)(22,30)(25,28)(26,32)(27,29);(1,3,5,4)(2,7,8,6)(9,11,14,13)(10,15,12,16)(17,20,22,31)(18,24,27,29)(19,21,30,23)(25,28,32,26);(1,4,5,3)(2,6,8,7)(9,12,14,10)(11,15,13,16)(17,20)(18,25)(19,21)(22,31)(23,30)(24,26)(27,32)(28,29);(9,13,14,11)(10,16,12,15)(17,21,22,23)(18,26,27,28)(19,20,30,31)(24,25,29,32)Polycyclic presentation:
2^9|b^2=fh,c^2=fhi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6533691);