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 | |
1931 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)(4,6)(5,8)(9,10)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,33)(24,32)(25,34)(26,36)(27,37)(28,38)(29,35)(30,40)(31,39);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,14)(13,27)(15,23)(16,25)(18,19)(20,37)(21,33)(22,34)(24,29)(26,30)(28,31)(32,35)(36,40)(38,39);(1,3,5,4)(2,7,8,6)(10,18)(11,23)(12,25)(14,15)(19,33)(20,35)(21,22)(24,27)(26,28)(29,31)(36,37)(39,40);(1,4,5,3)(2,6,8,7)(9,12,16,23)(10,19,22,33)(11,14,25,15)(13,28,30,24)(17,18,34,21)(20,38,40,32)(26,29,27,31)(35,37,39,36);(9,13,16,30)(10,20,22,40)(11,24,25,28)(12,26,23,27)(14,29,15,31)(17,32,34,38)(18,35,21,39)(19,36,33,37)Polycyclic presentation:
2^9|b^2=f,c^2=f,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, 6484793);