Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1443 |
| 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)(5,6,10,16,11,14,9,15)(7,17,20,28,21,12,19,29)(8,22,24,31,26,32,25,13)(18,30,23,27,34,36,35,33);(5,7)(6,12)(8,23,26,35)(9,20)(10,19)(11,21)(13,30,31,36)(14,17)(15,29)(16,28)(18,25,34,24)(22,33,32,27);(2,4)(5,8)(6,13,14,31)(7,18)(9,24)(10,25)(11,26)(12,27,17,33)(15,22,16,32)(19,23)(20,35)(21,34)(28,36,29,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,g^b=gi,f^c=fi,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, 31006);