Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353301 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,19)(11,20,23,29)(12,17,14,27)(13,18,25,33)(16,28,31,21)(22,34,36,39)(24,32,26,37)(30,38,40,35);(3,7)(5,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,35)(25,36)(26,38)(32,39)(33,40)(34,37);(1,2)(3,8)(4,6)(5,7)(9,12,15,14)(10,17,19,27)(11,21,23,28)(13,24,25,26)(16,29,31,20)(18,32,33,37)(22,35,36,38)(30,39,40,34);(9,13)(10,18)(11,22)(12,24)(14,26)(15,25)(16,30)(17,32)(19,33)(20,34)(21,35)(23,36)(27,37)(28,38)(29,39)(31,40);(13,25)(18,33)(22,36)(24,26)(30,40)(32,37)(34,39)(35,38)Polycyclic presentation:
2^9|a^2=h,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283125, #72283143, #72283150, … |
| maximal normal subgroup of | #72283125, #72283143, #72283150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260547);