Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,21)(12,16)(13,27)(14,18)(15,20)(17,23)(19,30)(22,35)(24,37)(25,38)(26,36)(28,31)(29,34)(32,39)(33,40);(1,3,4,8)(2,5,6,7)(10,16,20,18)(17,32,33,31)(19,27,36,34)(21,35,38,37);(9,11,15,25)(10,17,20,33)(12,24,14,22)(13,28,29,39)(16,31,18,32)(19,35,36,37)(21,34,38,27)(23,26,40,30);(2,6)(5,7)(10,16,20,18)(13,29)(17,31,33,32)(19,34,36,27)(21,35,38,37)(23,40)(26,30)(28,39)Polycyclic presentation:
2^9|b^2=egh,c^2=i,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415141);