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)(9,10,15,20,16,21,14,37)(11,22,26,17,27,39,25,30)(12,28,31,33,32,24,38,18)(13,19,35,29,36,40,34,23);(1,2)(3,7)(4,6)(5,8)(9,11,16,27)(10,17,21,30)(12,29,32,23)(13,33,36,18)(14,25,15,26)(19,31,40,38)(20,39,37,22)(24,34,28,35);(1,3,4,8)(2,5,6,7)(9,12,16,32)(10,18,21,33)(11,23,27,29)(13,17,36,30)(14,31,15,38)(19,26,40,25)(20,24,37,28)(22,34,39,35);(3,8)(5,7)(9,13,16,36)(10,19,21,40)(11,24,27,28)(12,30,32,17)(14,34,15,35)(18,25,33,26)(20,29,37,23)(22,38,39,31);(2,6)(5,7)(9,14,16,15)(10,20,21,37)(11,25,27,26)(12,31,32,38)(13,34,36,35)(17,39,30,22)(18,24,33,28)(19,29,40,23)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,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=df,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, 6970310);