Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574086 |
| 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 | #61216 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 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)(11,22)(12,17)(13,18)(14,19)(15,21)(16,25)(20,23)(24,39)(26,34)(27,35)(28,32)(29,36)(30,37)(31,38)(33,40);(2,6)(7,8)(9,10,14,19)(11,22,12,17)(13,28,30,38)(15,20,23,21)(16,25,34,26)(18,31,37,32)(24,33,27,36)(29,35,40,39);(1,3,5,4)(2,7,6,8)(9,11,23,26,14,12,15,25)(10,16,21,17,19,34,20,22)(13,27,32,40,30,24,31,29)(18,36,38,39,37,33,28,35);(1,4,5,3)(2,8,6,7)(9,12,14,11)(10,17,19,22)(13,29,30,40)(15,26,23,25)(16,21,34,20)(18,36,37,33)(24,32,27,31)(28,35,38,39);(9,13)(10,18)(11,24)(12,27)(14,30)(15,31)(16,33)(17,35)(19,37)(20,28)(21,38)(22,39)(23,32)(25,40)(26,29)(34,36)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,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, 6492008);