Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 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)(3,6)(4,5)(7,8)(9,10,15,20,16,21,14,30)(11,22,24,32,25,18,12,17)(13,19,29,35,27,34,28,39)(23,36,38,40,37,33,26,31);(2,5)(6,8)(9,11,14,12,16,25,15,24)(10,17,20,22,21,32,30,18)(13,23,28,26,27,37,29,38)(19,31,35,36,34,40,39,33);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,24,15,11)(10,18,20,17,21,22,30,32)(13,26,28,37,27,38,29,23)(19,33,35,31,34,36,39,40);(9,13,15,29,16,27,14,28)(10,19,20,35,21,34,30,39)(11,23,24,38,25,37,12,26)(17,31,22,36,32,40,18,33);(13,27)(19,34)(23,37)(26,38)(28,29)(31,40)(33,36)(35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280744);