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 | |
2068 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)(9,10,16,22)(11,23,25,31)(12,26,28,37)(13,19,17,32)(14,21,15,20)(18,33,35,29)(24,27,39,30)(34,38,36,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,22)(14,32)(15,19)(16,25)(18,27)(20,31)(21,23)(26,40)(28,39)(29,34)(30,35)(33,36)(37,38);(1,3,4,8)(2,5,6,7)(9,12,16,28)(10,18,22,35)(11,24,25,39)(13,30,17,27)(14,29,15,33)(19,36,32,34)(20,26,21,37)(23,38,31,40);(3,8)(5,7)(9,13)(10,19)(11,20)(12,27)(14,23)(15,31)(16,17)(18,34)(21,25)(22,32)(24,37)(26,39)(28,30)(29,40)(33,38)(35,36);(2,6)(5,7)(12,28)(18,35)(24,39)(26,37)(27,30)(29,33)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=i,c^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 6962163);