Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
873 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)(5,6,11,16)(7,18,22,19)(8,9,14,27)(10,15,24,26)(12,17,28,25)(13,29,31,21)(20,32,35,33)(23,30,36,34);(2,4)(5,7)(6,13)(8,19)(9,20,10,21)(11,22,28,36)(12,23)(14,30,15,18)(16,33,25,31)(17,32)(24,29)(26,34)(27,35);(5,8,24,6,12,26,27,17)(7,19,29,13,23,34,35,32)(9,25,28,15,10,16,11,14)(18,21,33,22,30,20,31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303790, #72313143, #72313146, … |
| maximal normal subgroup of | #72303790, #72313143, #72313146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54694);