Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1205 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)(5,6,9,14,10,15,17,12)(7,16,21,27,18,25,20,11)(8,13,29,28,24,32,23,31)(19,34,22,36,33,30,35,26);(5,7,17,20,10,18,9,21)(6,11,12,25,15,27,14,16)(8,22,29,33,24,35,23,19)(13,30,28,26,32,34,31,36);(6,12)(7,18)(9,17)(11,25)(13,28)(14,15)(16,27)(19,33)(23,29)(26,34)(30,36)(31,32);(2,4)(5,8,10,24)(6,13)(7,19)(9,23,17,29)(11,26,27,36)(12,28)(14,31)(15,32)(16,30,25,34)(18,33)(20,22)(21,35)Polycyclic presentation:
2^9|a^2=eg,b^2=fi,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 343904);