Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,10)(8,19,23,17)(9,24)(11,15)(12,16)(13,22,29,21)(14,25)(18,27)(20,33,36,28)(26,30)(31,34,32,35);(1,2,4,3)(5,7,11,16,12,15,10,6)(8,19,21,29,23,17,22,13)(9,18,26,30,27,24,25,14)(20,32,34,33,36,31,35,28);(5,8,11,22,12,23,10,21)(6,13,7,17,16,29,15,19)(9,20,26,35,27,36,25,34)(14,28,18,31,30,33,24,32);(1,3,4,2)(5,9,11,26,12,27,10,25)(6,14,7,18,16,30,15,24)(8,20,22,35,23,36,21,34)(13,28,17,31,29,33,19,32)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fg,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276387);