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 |
| prime radical | #60930 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,15,19,16,20,14,32)(11,21,23,17,12,24,26,34)(13,27,30,38,31,18,29,37)(22,35,40,36,25,39,28,33);(2,5)(3,7)(9,11)(10,17)(12,16)(13,28)(14,26)(15,23)(18,35)(19,24)(20,34)(21,32)(22,30)(25,29)(27,39)(31,40)(33,38)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12,16,11)(10,17,20,34)(13,22,31,25)(14,26,15,23)(18,36,27,33)(19,21,32,24)(28,29,40,30)(35,38,39,37);(2,5)(6,8)(9,13)(10,18)(11,22)(12,25)(14,29)(15,30)(16,31)(17,33)(19,37)(20,27)(21,39)(23,40)(24,35)(26,28)(32,38)(34,36);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,20,32)(11,23,12,26)(13,29,31,30)(17,21,34,24)(18,37,27,38)(22,40,25,28)(33,39,36,35)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,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, 7055307);