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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2572 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)(11,23)(12,26)(13,18)(14,19)(15,20)(16,22)(17,29)(21,36)(24,34)(25,32)(27,37)(28,40)(30,39)(31,33)(35,38);(1,3,5,4)(2,6,7,8)(9,11,15,26)(10,12,20,23)(13,24,31,19)(14,33,34,18)(16,17,36,27)(21,29,22,37)(25,35,40,30)(28,38,32,39);(9,12,15,23)(10,17,20,27)(11,22,26,21)(13,30,31,35)(14,28,34,32)(16,37,36,29)(18,19,33,24)(25,38,40,39);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,24)(12,14)(15,31)(16,28)(17,38)(19,26)(20,33)(21,25)(22,40)(23,34)(27,39)(29,35)(30,37)(32,36);(9,14,15,34)(10,19,20,24)(11,25,26,40)(12,28,23,32)(13,29,31,37)(16,35,36,30)(17,33,27,18)(21,39,22,38)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,e^2=i,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, 6484149);