Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
470 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)(5,6)(7,10)(8,22)(9,17)(11,16)(12,21)(13,19)(14,23)(15,28)(18,27)(20,25)(24,36)(26,34)(29,33)(30,32)(31,35);(5,7,11,21,12,15,9,6,13,17,28,18,27,16,10,19)(8,22,24,33,25,35,29,36,26,34,30,14,31,20,23,32);(5,8,9,23,27,31,11,30,13,26,10,29,12,25,28,24)(6,14,15,22,21,36,7,20,19,33,16,34,18,32,17,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250433, #72250442, #72250464, … |
| maximal normal subgroup of | #72250433, #72250442, #72250464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54134);