Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,16,9,15)(7,17,21,27,20,28,19,12)(8,13,24,32,23,30,25,31)(18,29,35,26,34,33,22,36);(1,2)(3,4)(5,7,9,19,11,20,10,21)(6,12,14,17,16,27,15,28)(8,22,24,18,23,35,25,34)(13,29,31,36,30,33,32,26);(2,4)(5,8)(6,13)(7,18,20,34)(9,25)(10,24)(11,23)(12,26,27,36)(14,32)(15,31)(16,30)(17,33,28,29)(19,22,21,35);(8,23)(13,30)(18,34)(22,35)(24,25)(26,36)(29,33)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252122, #72252125, #72252131, … |
| maximal normal subgroup of | #72252122, #72252125, #72252131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265675);