Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| 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×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1989 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)(3,8)(4,6)(5,7)(9,10,16,22)(11,17,26,37)(12,27,30,31)(13,19,33,40)(14,20,15,21)(18,34,38,29)(23,32,39,35)(24,28,25,36);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,19)(13,31)(14,25)(15,24)(16,26)(18,32)(20,36)(21,28)(22,37)(23,29)(27,33)(30,40)(34,39)(35,38);(9,12,14,29,16,30,15,34)(10,18,21,27,22,38,20,31)(11,23,25,40,26,39,24,19)(13,28,32,37,33,36,35,17);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,18)(12,28)(14,32)(15,35)(16,33)(17,34)(20,39)(21,23)(22,40)(24,31)(25,27)(26,38)(29,37)(30,36);(9,14,16,15)(10,20,22,21)(11,24,26,25)(12,29,30,34)(13,32,33,35)(17,28,37,36)(18,31,38,27)(19,39,40,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351162);