Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,8,22,15)(7,16,19,33,21,34,20,30)(9,24,26,13,27,31,36,23)(12,25,28,32,17,35,29,18);(5,7,10,19,11,21,22,20)(6,12,15,29,8,17,14,28)(9,25,26,32,27,35,36,18)(13,30,24,34,23,33,31,16);(2,4)(5,8)(6,11)(7,17)(9,23)(10,14)(12,21)(13,27)(15,22)(16,32)(18,34)(19,28)(20,29)(24,36)(25,33)(26,31)(30,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321300, #72321366, #72321379, … |
| maximal normal subgroup of | #72321300, #72321366, #72321379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53754);