Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461151 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16381 |
| 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 | |
883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,16,26,18,25,17,8)(4,19,21,9,22,28,20,27)(14,23,15,24,32,30,31,29);(1,3,6,16,7,18,5,17)(2,8,11,13,12,26,10,25)(4,15,20,32,22,31,21,14)(9,24,19,30,27,29,28,23);(1,4,5,20,7,22,6,21)(2,9,10,19,12,27,11,28)(3,14,16,31,18,32,17,15)(8,23,13,29,26,30,25,24);(1,4)(2,9)(3,15)(5,21)(6,20)(7,22)(8,24)(10,28)(11,19)(12,27)(13,30)(14,17)(16,32)(18,31)(23,25)(26,29)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368347);