Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478231 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17136 |
| 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 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,23)(3,9,16,27,17,29,24,28)(4,18,5,11,20,32,21,10)(14,31,19,26,22,30,15,25);(1,3,6,16,8,17,7,24)(2,9,12,27,13,29,23,28)(4,19,21,14,20,15,5,22)(10,30,32,26,11,31,18,25);(1,4,6,21,8,20,7,5)(2,10,12,32,13,11,23,18)(3,14,16,15,17,22,24,19)(9,25,27,30,29,26,28,31);(1,5,6,20,8,21,7,4)(2,11,12,32,13,10,23,18)(3,15,16,14,17,19,24,22)(9,26,27,30,29,25,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=ef,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385427);