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 | |
2217 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)(9,10,14,19,15,20,13,18)(11,21,23,31,25,36,24,16)(12,17,28,34,29,35,27,33)(22,32,26,39,38,40,37,30);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,32)(18,21)(19,36)(20,31)(22,28)(27,38)(29,37)(30,34)(33,39)(35,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,23)(22,30)(24,36)(25,31)(26,32)(27,33)(28,34)(29,35)(37,40)(38,39);(3,8)(5,7)(9,12,14,28,15,29,13,27)(10,17,19,34,20,35,18,33)(11,22,23,26,25,38,24,37)(16,30,21,32,31,39,36,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,36)(17,33,35,34)(22,26,38,37)(30,32,39,40)Polycyclic presentation:
2^9|a^2=f,d^2=f,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, 6903597);