Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,13,19,15,20,14,18)(11,21,23,16,25,38,24,33)(12,26,29,36,31,17,30,37)(22,39,40,35,28,34,27,32);(2,5)(3,7)(9,11,15,25)(10,16,20,33)(12,27,31,40)(13,24,14,23)(17,34,26,39)(18,38,19,21)(22,29,28,30)(32,37,35,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,28)(13,23)(14,24)(15,25)(17,35)(18,21)(19,38)(20,33)(22,31)(26,32)(27,29)(30,40)(34,37)(36,39);(2,5)(6,8)(9,12,13,29,15,31,14,30)(10,17,19,37,20,26,18,36)(11,22,23,40,25,28,24,27)(16,32,38,39,33,35,21,34);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,29,31,30)(16,21,33,38)(17,36,26,37)(22,40,28,27)(32,34,35,39)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053030);