Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | |
1370 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)(5,6,10,16,11,12,9,15)(7,17,18,26,20,28,25,13)(8,21,22,14,23,30,24,27)(19,29,32,31,34,35,33,36);(6,12)(7,18)(9,10)(13,28)(17,26)(19,33)(20,25)(21,30)(22,24)(29,36)(31,35)(32,34);(2,4)(5,7,10,25,11,20,9,18)(6,13,15,17,12,26,16,28)(8,19,22,33,23,34,24,32)(14,29,21,31,27,35,30,36);(2,4)(5,8)(6,14)(7,19)(9,24)(10,22)(11,23)(12,27)(13,29)(15,21)(16,30)(17,31)(18,32)(20,34)(25,33)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295568);