Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19,16,20,14,31)(11,21,23,17,12,24,26,33)(13,18,29,36,30,38,28,37)(22,34,27,32,25,40,39,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,27)(14,26)(15,23)(18,34)(19,24)(20,33)(21,31)(22,28)(25,29)(30,39)(32,36)(35,37)(38,40);(1,3,4,8)(2,5,6,7)(9,12,16,11)(10,17,20,33)(13,22,30,25)(14,26,15,23)(18,35,38,32)(19,21,31,24)(27,29,39,28)(34,36,40,37);(3,8)(5,7)(9,13)(10,18)(11,22)(12,25)(14,28)(15,29)(16,30)(17,32)(19,36)(20,38)(21,34)(23,27)(24,40)(26,39)(31,37)(33,35);(2,6)(5,7)(9,14,16,15)(10,19,20,31)(11,23,12,26)(13,28,30,29)(17,21,33,24)(18,36,38,37)(22,27,25,39)(32,34,35,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970316);