Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #399126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,16)(9,10)(13,17)(14,29)(15,24)(21,23)(25,28)(26,30)(27,32)(33,34)(35,36);(1,2)(3,4)(5,6,9,12,11,15,10,24)(7,17,20,13,16,25,19,28)(8,14,23,26,22,30,21,29)(18,32,33,27,31,35,34,36);(2,4)(5,7,9,20,11,16,10,19)(6,13,12,25,15,28,24,17)(8,18,23,33,22,31,21,34)(14,27,26,35,30,36,29,32);(5,8)(6,14)(7,18)(9,23)(10,21)(11,22)(12,26)(13,27)(15,30)(16,31)(17,32)(19,34)(20,33)(24,29)(25,35)(28,36)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306322);