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)(9,10,14,31,16,20,15,19)(11,21,24,17,25,33,23,13)(12,26,34,18,30,35,29,36)(22,28,38,37,27,32,39,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,24,16,25,15,23)(10,17,19,21,20,13,31,33)(12,27,29,38,30,22,34,39)(18,32,35,40,36,28,26,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,22)(13,32)(14,29)(15,34)(16,30)(17,28)(19,35)(20,36)(21,37)(23,38)(24,39)(25,27)(26,31)(33,40);(3,8)(5,7)(9,13)(10,11)(12,28)(14,21)(15,33)(16,17)(18,27)(19,23)(20,25)(22,36)(24,31)(26,38)(29,40)(30,32)(34,37)(35,39);(2,6)(5,7)(9,14,16,15)(10,19,20,31)(11,23,25,24)(12,29,30,34)(13,21,17,33)(18,35,36,26)(22,38,27,39)(28,40,32,37)Polycyclic presentation:
2^9|a^2=gi,b^2=f,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, 6910270);