Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,18)(14,16)(17,27)(19,35)(20,31)(21,40)(22,33)(24,30)(25,37)(26,36)(28,38)(29,34)(32,39);(2,5)(6,8)(9,11,13,28,14,20,29,22)(10,15,18,38,16,31,34,33)(12,24,26,21,27,19,25,39)(17,35,37,32,23,30,36,40);(11,19)(12,25)(13,29)(15,30)(17,36)(18,34)(20,24)(21,28)(22,39)(23,37)(26,27)(31,35)(32,38)(33,40);(1,3,4,7)(2,6,5,8)(10,16)(15,31)(17,23)(18,34)(30,35)(32,40)(33,38)(36,37)Polycyclic presentation:
2^9|b^2=fh,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237246, #72239821, #72239836, … |
| maximal normal subgroup of | #72237246, #72239821, #72239836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406789);