Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | #60376 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2601 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)(11,17)(12,28)(13,19)(14,20)(15,22)(16,23)(18,25)(21,33)(24,36)(26,39)(27,35)(29,37)(30,40)(31,38)(32,34);(1,3,5,4)(2,6,7,8)(9,11,16,13)(10,17,23,19)(12,14,29,32)(15,26,33,27)(18,36,38,40)(20,37,34,28)(21,35,22,39)(24,31,30,25);(9,12,16,29)(10,18,23,38)(11,24,13,30)(14,27,32,26)(15,31,33,25)(17,34,19,20)(21,37,22,28)(35,40,39,36);(1,4,5,3)(2,6,7,8)(9,13,16,11)(10,19,23,17)(12,14,29,32)(15,27,33,26)(18,36,38,40)(20,37,34,28)(21,39,22,35)(24,31,30,25);(9,14)(10,20)(11,25)(12,27)(13,31)(15,24)(16,32)(17,18)(19,38)(21,40)(22,36)(23,34)(26,29)(28,35)(30,33)(37,39)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=fi,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,g^c=gi,h^c=hi,e^d=ehi,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, 6484156);