Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219542 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,14,11,16,9,15)(7,17,19,29,21,28,20,12)(8,22,25,31,26,32,24,13)(18,27,34,30,35,36,23,33);(1,2)(3,4)(5,7,9,20,11,21,10,19)(6,12,14,17,16,29,15,28)(8,23,24,35,26,34,25,18)(13,30,22,36,31,33,32,27);(2,4)(5,8)(6,13,16,31)(7,18)(9,25)(10,24)(11,26)(12,27,29,36)(14,32,15,22)(17,33,28,30)(19,23)(20,34)(21,35);(7,19)(9,10)(12,28)(14,15)(17,29)(18,23)(20,21)(22,32)(24,25)(27,30)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295648);