Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,17,29,33,9,25,13,15,10,16,12,18,27,8)(7,19,21,31,30,36,28,32,23,34,26,14,22,24,20,35);(1,2)(3,4)(5,7)(6,14,15,31)(8,24,25,36)(9,26)(10,28)(11,20)(12,30)(13,23)(16,19,17,34)(18,35,33,32)(21,27)(22,29);(2,4)(5,8,9,18,12,16,11,15,13,25,27,33,29,17,10,6)(7,14,20,19,22,32,26,24,23,31,28,34,30,35,21,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,f^b=fhi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56264);