Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,9,15,25,18)(7,19,23,33,22,30,24,34)(8,17,27,14,26,13,11,28)(12,29,32,21,31,20,35,36);(5,7,11,20,25,23,26,36,9,22,27,29,10,24,8,21)(6,12,17,33,16,35,14,19,15,31,13,34,18,32,28,30);(1,2,3,4)(5,8,25,11,9,26,10,27)(6,13,18,14,15,17,16,28)(7,20,24,21,22,29,23,36)(12,30,35,33,31,19,32,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250629, #72250647, #72250653, … |
| maximal normal subgroup of | #72250629, #72250647, #72250653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54627);