Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | |
2592 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)(11,18)(12,28)(13,20)(14,21)(15,33)(16,22)(17,23)(19,30)(24,40)(25,35)(26,36)(27,37)(29,34)(31,38)(32,39);(1,3,5,4)(2,6,7,8)(9,11)(10,18)(12,29)(13,16)(14,30)(15,26)(17,27)(19,21)(20,22)(23,37)(24,31)(25,32)(28,34)(33,36)(35,39)(38,40);(9,12,17,25)(10,19,23,38)(11,24,27,14)(13,29,26,32)(15,31,16,30)(18,34,37,39)(20,21,36,40)(22,35,33,28);(1,4,5,3)(2,6,7,8)(9,13,17,26)(10,20,23,36)(11,16,27,15)(12,24,25,14)(18,22,37,33)(19,34,38,39)(21,28,40,35)(29,31,32,30);(9,14,17,24)(10,21,23,40)(11,25,27,12)(13,30,26,31)(15,32,16,29)(18,35,37,28)(19,36,38,20)(22,34,33,39)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=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, 6483931);