Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,16,21,15,32)(11,22,24,13,19,31,25,17)(12,26,29,36,30,35,28,18)(23,33,39,40,34,27,38,37);(1,2)(3,7)(4,6)(5,8)(9,11,14,24,16,19,15,25)(10,17,32,31,21,13,20,22)(12,23,28,38,30,34,29,39)(18,33,26,40,36,27,35,37);(1,3)(2,5)(4,8)(6,7)(9,12,16,30)(10,18,21,36)(11,23,19,34)(13,27,17,33)(14,28,15,29)(20,35,32,26)(22,37,31,40)(24,38,25,39);(3,8)(5,7)(9,13)(10,19)(11,21)(12,27)(14,31)(15,22)(16,17)(18,34)(20,25)(23,36)(24,32)(26,38)(28,40)(29,37)(30,33)(35,39);(2,6)(5,7)(9,14,16,15)(10,20,21,32)(11,24,19,25)(12,28,30,29)(13,31,17,22)(18,35,36,26)(23,38,34,39)(27,40,33,37)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^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=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910274);