Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196257 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,19,13,18)(11,16,22,30,23,31,28,35)(12,24,26,17,27,34,25,33)(21,36,37,32,38,40,39,29);(2,5)(6,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,32)(18,35)(19,31)(20,30)(24,36)(25,39)(26,37)(27,38)(29,33)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,27)(10,17,19,33)(11,21,23,38)(13,26,14,25)(16,29,31,32)(18,34,20,24)(22,39,28,37)(30,40,35,36);(3,7)(6,8)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286959);