Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219706 |
| 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:
(5,6,11,15,10,16,9,14)(7,17,19,28,21,29,20,12)(8,22,25,32,26,13,24,31)(18,33,23,36,35,30,34,27);(1,2)(3,4)(5,7,9,20,10,21,11,19)(6,12,15,17,16,28,14,29)(8,23,24,18,26,34,25,35)(13,30,31,27,22,33,32,36);(2,4)(5,8)(6,13,16,22)(7,18)(9,25)(10,26)(11,24)(12,27,28,36)(14,31,15,32)(17,30,29,33)(19,34)(20,23)(21,35);(6,14)(7,19)(9,11)(12,28)(13,31)(15,16)(18,34)(20,21)(22,32)(23,35)(24,25)(27,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 295658);