Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
831 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)(5,6,10,14,11,15,8,21)(7,16,17,27,20,33,18,12)(9,13,22,29,23,30,24,28)(19,32,35,26,36,25,34,31);(1,2,3,4)(5,7,8,18,11,20,10,17)(6,12,14,16,15,27,21,33)(9,19,22,35,23,36,24,34)(13,26,28,32,30,31,29,25);(6,13)(7,17)(8,10)(12,25)(14,29)(15,30)(16,31)(18,20)(19,34)(21,28)(22,24)(26,33)(27,32)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^c=di,f^c=fi,g^c=gh,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312687, #72313143, #72313146, … |
| maximal normal subgroup of | #72312687, #72313143, #72313146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55217);