Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10,15,20,16,22,13,17)(11,23,24,39,28,38,27,35)(12,29,14,32,30,21,31,19)(18,34,33,26,37,25,36,40);(1,2)(3,6)(4,5)(7,8)(10,17,22,20)(11,24,28,27)(12,30)(14,31)(19,29,32,21)(23,38)(25,26,34,40)(35,39);(2,5)(6,8)(9,11,14,26,15,27,30,34,16,28,31,40,13,24,12,25)(10,18,21,23,17,33,32,39,22,37,29,38,20,36,19,35);(1,3,4,7)(2,6,5,8)(9,12,13,31,16,30,15,14)(10,19,20,29,22,32,17,21)(11,25,24,40,28,34,27,26)(18,35,36,38,37,39,33,23)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=f,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309856, #72309864, #72309865, … |
| maximal normal subgroup of | #72309856, #72309864, #72309865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404254);