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 | |
1780 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)(9,10,14,20,15,21,13,19)(11,22,24,16,25,32,30,17)(12,18,28,36,29,37,27,35)(23,38,39,31,40,34,26,33);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,30)(14,24)(15,25)(17,21)(18,34)(19,22)(20,32)(23,28)(27,40)(29,39)(31,35)(33,36)(37,38);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,23)(13,24)(14,30)(15,25)(16,21)(18,33)(19,22)(20,32)(26,28)(27,39)(29,40)(31,37)(34,36)(35,38);(3,8)(5,7)(9,12,15,29)(10,18,21,37)(11,23,25,40)(13,27,14,28)(16,31,17,33)(19,35,20,36)(22,38,32,34)(24,39,30,26);(2,6)(5,7)(9,13,15,14)(10,19,21,20)(11,24,25,30)(12,27,29,28)(16,32,17,22)(18,35,37,36)(23,39,40,26)(31,34,33,38)Polycyclic presentation:
2^9|a^2=g,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, 6970302);