Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,12,9,11,16)(7,17,19,26,20,29,27,13)(8,21,23,14,25,31,24,22)(18,28,33,36,35,30,34,32);(1,2)(3,4)(5,7,11,19,12,20,10,27)(6,13,16,17,9,26,15,29)(8,18,24,33,25,35,23,34)(14,30,21,32,22,28,31,36);(2,4)(5,8)(6,14)(7,18,20,35)(9,22)(10,24)(11,23)(12,25)(13,28,26,30)(15,21)(16,31)(17,32,29,36)(19,34,27,33);(5,9,12,6)(7,17)(8,22,25,14)(10,16,11,15)(13,27)(18,32)(19,26)(20,29)(21,24,31,23)(28,33)(30,34)(35,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ef,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339205);