Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19,16,12,14,27)(11,20,24,21,25,36,23,17)(13,18,30,34,31,26,29,35)(22,33,38,37,39,40,28,32);(1,2)(3,7)(4,6)(5,8)(9,11,14,23,16,25,15,24)(10,17,19,20,12,21,27,36)(13,28,30,22,31,38,29,39)(18,33,35,32,26,40,34,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,21)(13,26)(14,27)(15,19)(17,25)(18,31)(20,24)(22,37)(23,36)(28,40)(29,35)(30,34)(32,39)(33,38);(3,8)(5,7)(9,13)(10,18)(11,22)(12,26)(14,29)(15,30)(16,31)(17,32)(19,34)(20,33)(21,37)(23,28)(24,38)(25,39)(27,35)(36,40);(2,6)(5,7)(9,14,16,15)(10,19,12,27)(11,23,25,24)(13,29,31,30)(17,20,21,36)(18,34,26,35)(22,28,39,38)(32,33,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910287);