Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140370 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1943 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,13)(7,17,20,36)(8,11)(9,22)(10,21)(14,30,32,35)(15,34)(16,27)(18,29,31,33)(19,26,28,24);(5,6,8,13,12,16,11,27)(7,18,19,35,20,31,28,30)(9,23,21,34,10,25,22,15)(14,24,29,36,32,26,33,17);(2,4)(5,7,8,19,12,20,11,28)(6,14,13,29,16,32,27,33)(9,24,21,36,10,26,22,17)(15,35,23,31,34,30,25,18)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250508, #72250510, #72250519, … |
| maximal normal subgroup of | #72250508, #72250510, #72250519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55965);