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 | |
1761 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)(9,10,15,20,16,21,14,36)(11,22,26,17,27,38,25,32)(12,28,30,39,31,24,37,18)(13,19,34,23,35,40,33,29);(1,2)(3,7)(4,6)(5,8)(9,11,16,27)(10,17,21,32)(12,29,31,23)(13,18,35,39)(14,25,15,26)(19,37,40,30)(20,38,36,22)(24,34,28,33);(1,3,4,8)(2,5,6,7)(9,12,16,31)(10,18,21,39)(11,23,27,29)(13,32,35,17)(14,30,15,37)(19,25,40,26)(20,24,36,28)(22,34,38,33);(3,8)(5,7)(9,13)(10,19)(11,24)(12,17)(14,33)(15,34)(16,35)(18,26)(20,23)(21,40)(22,37)(25,39)(27,28)(29,36)(30,38)(31,32);(2,6)(5,7)(9,14,16,15)(10,20,21,36)(11,25,27,26)(12,30,31,37)(13,33,35,34)(17,38,32,22)(18,24,39,28)(19,23,40,29)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,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, 6970340);