Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1030 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)(5,6,10,17,12,18,11,29)(7,19,23,30,20,35,28,31)(8,15,26,14,25,34,9,16)(13,22,32,24,33,27,36,21);(2,4)(5,7,11,23,12,20,10,28)(6,13,17,32,18,33,29,36)(8,24,26,22,25,21,9,27)(14,31,15,35,16,30,34,19);(6,14)(7,20)(8,25)(9,26)(13,30)(15,17)(16,18)(19,32)(23,28)(29,34)(31,33)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53911);