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 | |
2233 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)(9,10,14,19,16,20,15,31)(11,21,23,33,12,24,26,17)(13,18,28,35,30,36,29,40)(22,34,27,39,25,37,38,32);(1,2)(3,7)(4,6)(5,8)(9,11,16,12)(10,17,20,33)(13,27,30,38)(14,26,15,23)(18,34,36,37)(19,24,31,21)(22,29,25,28)(32,40,39,35);(1,3)(2,5)(4,8)(6,7)(9,12,16,11)(10,17,20,33)(13,25,30,22)(14,26,15,23)(18,32,36,39)(19,21,31,24)(27,28,38,29)(34,40,37,35);(3,8)(5,7)(9,13,15,29,16,30,14,28)(10,18,31,40,20,36,19,35)(11,22,26,38,12,25,23,27)(17,32,24,37,33,39,21,34);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,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, 6903596);