Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17)(8,14,22,29)(9,12,10,15)(13,24)(18,31)(19,28)(20,27)(21,30,23,25)(26,32)(33,35)(34,36);(1,2)(3,4)(6,12,16,15)(8,21,22,23)(13,17,28,27)(14,29)(18,32,34,33)(25,30)(26,35)(31,36);(5,7,9,19,11,20,10,24)(6,13,12,17,16,28,15,27)(8,18,23,33,22,34,21,32)(14,26,25,31,29,35,30,36);(2,4)(5,8,9,23,11,22,10,21)(6,14,12,25,16,29,15,30)(7,18,19,33,20,34,24,32)(13,26,17,31,28,35,27,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,f^c=fi,g^c=gi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264189);