Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574075 |
| 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 | #61288 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 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)(12,18)(13,19)(14,21)(15,23)(16,24)(20,31)(22,35)(25,34)(26,40)(27,33)(28,36)(29,38)(30,37)(32,39);(2,6)(7,8)(9,11,13,23)(10,15,19,17)(12,26,28,27)(14,24,31,25)(16,21,34,20)(18,33,36,40)(22,30,39,29)(32,37,35,38);(9,11,14,24,13,23,31,25)(10,16,20,17,19,34,21,15)(12,22,29,27,28,39,30,26)(18,33,37,35,36,40,38,32);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,27)(13,23)(14,24)(15,19)(16,21)(18,33)(20,34)(22,30)(25,31)(26,28)(29,39)(32,38)(35,37)(36,40);(1,4,5,3)(2,8,6,7)(9,12,13,28)(10,18,19,36)(11,22,23,39)(14,29,31,30)(15,32,17,35)(16,33,34,40)(20,37,21,38)(24,27,25,26)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492257);