Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #402446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| 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 | |
957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,17,29,15)(7,18,20,12,22,34,21,30)(8,23,36,13,26,31,25,32)(9,24,27,14,28,19,35,33);(5,7,10,20,11,22,29,21)(6,12,15,18,17,30,16,34)(8,24,25,33,26,19,36,14)(9,13,27,31,28,32,35,23);(5,8,11,26)(6,13)(7,19)(9,18,28,34)(10,25,29,36)(12,27,30,35)(14,20)(15,31)(16,23)(17,32)(21,33)(22,24);(2,4)(5,9,10,27,11,28,29,35)(6,14,15,24,17,33,16,19)(7,13,20,31,22,32,21,23)(8,18,25,30,26,34,36,12)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309642);