Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(9,10,15,20)(11,21,25,30)(12,26,29,33)(13,18,14,19)(16,22,32,24)(17,27,34,28)(23,31,38,40)(35,37,39,36);(1,2,4,3)(5,6)(7,8)(11,22)(12,27)(13,14)(16,30)(17,33)(18,19)(21,32)(24,25)(26,34)(28,29)(35,39)(36,37);(9,11,15,25)(10,16,20,32)(12,23,29,38)(13,22,14,24)(17,31,34,40)(18,21,19,30)(26,35,33,39)(27,36,28,37);(6,8)(9,12,15,29)(10,17,20,34)(11,23,25,38)(13,27,14,28)(16,31,32,40)(18,26,19,33)(21,35,30,39)(22,36,24,37);(1,3,4,2)(9,13,15,14)(10,18,20,19)(11,24,25,22)(12,27,29,28)(16,30,32,21)(17,26,34,33)(23,37,38,36)(31,39,40,35)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426478);