Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
2245 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)(9,10,14,21,16,22,15,20)(11,23,25,37,26,36,29,17)(12,27,30,18,31,39,33,35)(13,19,28,34,32,40,24,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,28)(13,30)(14,29)(15,25)(16,26)(18,38)(19,27)(20,23)(21,36)(22,37)(24,31)(32,33)(34,35)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,25)(14,30)(15,33)(16,31)(17,34)(19,36)(20,27)(21,39)(22,35)(23,40)(26,28)(29,32)(37,38);(3,8)(5,7)(9,13,15,24,16,32,14,28)(10,19,20,38,22,40,21,34)(11,12,29,33,26,31,25,30)(17,35,36,39,37,18,23,27);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,25,26,29)(12,30,31,33)(13,28,32,24)(17,36,37,23)(18,27,35,39)(19,38,40,34)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903599);