Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,22,24)(19,27,29,32)(20,26,21,25)(23,30,31,28);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(18,24)(23,31)(25,26)(27,32);(10,13)(14,16)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402053);