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 | |
1191 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)(5,6,10,19,12,17,11,16)(7,18,29,8,21,23,14,13)(9,24,27,33,28,15,26,32)(20,31,25,30,34,35,36,22);(5,7,10,29,12,21,11,14)(6,13,19,18,17,8,16,23)(9,25,26,20,28,36,27,34)(15,31,33,22,24,35,32,30);(5,8,12,13)(6,14,17,29)(7,19,21,16)(9,22,28,30)(10,18,11,23)(15,25,24,36)(20,33,34,32)(26,31,27,35);(2,4)(5,9)(6,15,17,24)(7,20,21,34)(8,22)(10,26)(11,27)(12,28)(13,30)(14,25,29,36)(16,32,19,33)(18,31)(23,35)Polycyclic presentation:
2^9|a^2=eg,b^2=fi,c^2=i,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, 343905);