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 | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,12,9,11,26)(7,16,18,25,20,29,19,13)(8,21,23,22,24,31,27,14)(17,30,33,32,35,28,34,36);(1,2)(3,4)(5,7,10,19,12,20,11,18)(6,13,15,29,9,25,26,16)(8,17,23,34,24,35,27,33)(14,30,21,36,22,28,31,32);(2,4)(5,8)(6,14)(7,17,20,35)(9,22)(10,27)(11,23)(12,24)(13,28,25,30)(15,31)(16,32,29,36)(18,34,19,33)(21,26);(5,9,12,6)(7,16)(8,22,24,14)(10,26,11,15)(13,19)(17,32)(18,25)(20,29)(21,23,31,27)(28,33)(30,34)(35,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=efi,d^2=ef,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, 339206);