Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C24 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,23)(9,26)(10,25)(11,15)(12,16)(13,19)(14,24)(18,21)(20,30)(22,32)(27,28)(29,36)(31,34)(33,35);(5,7,10,21,11,22,28,31,12,19,23,35,9,20,14,29)(6,13,8,18,15,30,24,34,16,17,25,33,26,32,27,36);(1,3,4,2)(5,8)(6,14)(7,18)(9,27)(10,15)(11,24)(12,25)(13,29)(16,28)(17,31)(19,33)(20,36)(21,30)(22,34)(23,26)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,d^b=di,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54319);