Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533141 |
| 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 | C24 | C25 |
| prime radical | #62900 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,19,22,15)(12,23,25,16)(13,26)(14,18)(17,27)(20,31)(21,29,35,32)(24,30,37,36)(28,34)(33,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,18,19)(12,20,25,34)(13,21,27,35)(16,28,23,31)(17,29,26,32)(24,33,37,40)(30,38,36,39);(2,5)(6,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,31)(21,33)(22,34)(26,36)(27,37)(29,38)(32,39)(35,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,27)(10,17,18,26)(11,21,22,35)(12,24,25,37)(15,29,19,32)(16,30,23,36)(20,33,34,40)(28,38,31,39);(3,7)(6,8)(10,18)(15,19)(16,23)(17,26)(28,31)(29,32)(30,36)(38,39)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449443);