Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563333 |
| 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 |
| prime radical | C2×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,14,19)(11,20,23,15)(12,25,29,40)(13,30,31,18)(16,34,37,28)(17,22,33,24)(21,36,27,35)(26,38,39,32);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,26)(13,22)(14,23)(16,35)(17,30)(18,33)(19,20)(21,28)(24,31)(25,32)(27,34)(29,39)(36,37)(38,40);(9,12,13,28,14,29,31,34)(10,16,18,25,19,37,30,40)(11,21,24,26,23,27,22,39)(15,32,17,35,20,38,33,36);(9,11)(10,17,19,33)(12,27)(13,24)(14,23)(15,18,20,30)(16,36,37,35)(21,29)(22,31)(25,32,40,38)(26,34)(28,39);(1,4,5,3)(2,7,8,6)(9,13,14,31)(10,18,19,30)(11,22,23,24)(12,28,29,34)(15,33,20,17)(16,25,37,40)(21,39,27,26)(32,36,38,35)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=h,e^2=f,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,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521322);