Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15,10,9,12,25)(7,16,18,29,19,13,26,28)(8,20,23,32,22,14,24,31)(17,27,21,30,34,33,35,36);(5,7)(6,13,9,16)(8,21)(10,19)(11,26)(12,18)(14,30,20,36)(15,29,25,28)(17,23)(22,35)(24,34)(27,32,33,31);(5,8,10,22)(6,14,9,20)(7,17,19,34)(11,24,12,23)(13,27,16,33)(15,32,25,31)(18,35,26,21)(28,36,29,30);(2,4)(5,9,10,6)(7,13)(8,20,22,14)(11,25,12,15)(16,19)(17,27)(18,28)(21,30)(23,32,24,31)(26,29)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343833);