Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16562 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,6,10,22)(4,12,17,24)(5,13)(8,15,25,16)(9,20)(14,23)(18,32)(19,28)(21,27)(26,30)(29,31);(1,3,6,11,7,10,22,2)(4,8,15,24,17,25,16,12)(5,13,20,28,21,27,19,9)(14,23,30,32,31,29,18,26);(1,4,6,16,7,17,22,15)(2,8,3,12,11,25,10,24)(5,18,20,31,21,30,19,14)(9,26,13,29,28,32,27,23);(1,5)(2,9)(3,13)(4,14)(6,20)(7,21)(8,23)(10,27)(11,28)(12,26)(15,30)(16,18)(17,31)(19,22)(24,32)(25,29)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374438);