Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| 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 | |
1434 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)(7,17,20,12)(8,22,25,32)(9,15)(10,16)(11,14)(13,24,31,26)(18,33)(19,29,21,28)(23,36)(27,34)(30,35);(5,7,10,21,11,20,9,19)(6,12,16,29,14,17,15,28)(8,23,24,35,25,18,26,34)(13,30,22,33,31,27,32,36);(2,4)(5,8)(6,13,14,31)(7,18)(9,24)(10,26)(11,25)(12,27,17,30)(15,22,16,32)(19,34)(20,23)(21,35)(28,36,29,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30957);