Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470362 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1532 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)(19,26,27,32)(20,25,21,30)(23,28,31,29);(9,10)(11,14)(12,13)(15,16)(17,19,21,29,22,27,20,28)(18,23,25,26,24,31,30,32);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(18,24)(19,27)(25,30)(28,29);(10,13)(14,16)(17,20,22,21)(18,25,24,30)(19,28,27,29)(23,26,31,32);(9,11,12,15)(10,14,13,16)(18,24)(19,27)(25,30)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402223);