Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
2072 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,24)(13,30)(14,20)(15,33)(16,21)(18,28)(19,31)(22,26)(23,32)(25,34)(27,36)(29,38)(35,40)(37,39);(2,6)(7,8)(9,11,16,19)(10,17,21,31)(12,25,29,37)(13,20,32,33)(14,23,15,30)(18,26,36,40)(22,27,35,28)(24,34,38,39);(1,3,5,4)(2,7,6,8)(9,12,15,28,16,29,14,27)(10,18,20,24,21,36,33,38)(11,22,23,25,19,35,30,37)(13,26,31,39,32,40,17,34);(1,4,5,3)(2,8,6,7)(9,13,14,17,16,32,15,31)(10,19,33,23,21,11,20,30)(12,26,27,34,29,40,28,39)(18,35,38,25,36,22,24,37);(2,6)(7,8)(9,14,16,15)(10,20,21,33)(11,23,19,30)(12,27,29,28)(13,17,32,31)(18,24,36,38)(22,25,35,37)(26,34,40,39)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468974);