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 | |
1766 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,5)(6,8)(10,13)(14,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,25)(23,32)(24,29)(26,31);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,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, 6402415);