Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1986 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,24)(13,19)(14,20)(15,31)(16,21)(18,23)(22,32)(25,40)(26,33)(27,35)(28,36)(29,37)(30,38)(34,39);(1,3,5,4)(2,6,7,8)(9,11,16,23)(10,17,21,18)(12,14,26,15)(13,27,30,25)(19,35,38,40)(20,33,31,24)(22,28,39,29)(32,36,34,37);(9,12,15,23,16,26,14,11)(10,18,20,24,21,17,31,33)(13,25,29,39,30,27,28,22)(19,34,36,40,38,32,37,35);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,22)(12,25)(14,28)(15,29)(16,30)(17,32)(18,34)(20,36)(21,38)(23,39)(24,40)(26,27)(31,37)(33,35);(9,14,16,15)(10,20,21,31)(11,12,23,26)(13,28,30,29)(17,24,18,33)(19,36,38,37)(22,25,39,27)(32,40,34,35)Polycyclic presentation:
2^9|b^2=fi,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,e^c=ei,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, 6351171);