Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1590 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)(7,16,19,12)(8,21)(9,25)(10,15)(11,14)(13,23)(17,31,32,36)(18,27,20,30)(22,35)(24,29)(26,33,28,34);(5,7,10,20,11,19,9,18)(6,12,15,27,14,16,25,30)(8,17,24,34,22,32,23,33)(13,28,21,31,29,26,35,36);(1,2,3,4)(5,8,9,23,11,22,10,24)(6,13,15,21,14,29,25,35)(7,17,18,33,19,32,20,34)(12,26,27,36,16,28,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31093);