Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,18,12,16,10,17)(7,19,22,29,8,13,21,14)(9,24,25,15,26,33,27,32)(20,30,34,31,23,36,35,28);(1,2)(3,4)(5,7,12,8)(6,13,16,19)(9,20,26,23)(10,22,11,21)(14,18,29,17)(15,31,32,28)(24,36,33,30)(25,35,27,34);(2,4)(5,8,12,7)(6,14,16,29)(9,23,26,20)(10,21,11,22)(13,17,19,18)(15,30,32,36)(24,31,33,28)(25,34,27,35);(5,9)(6,15)(7,20)(8,23)(10,27)(11,25)(12,26)(13,28)(14,30)(16,32)(17,24)(18,33)(19,31)(21,35)(22,34)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,f^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,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290355);