Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1554 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)(5,6,9,14,11,15,10,16)(7,17,19,26,20,30,23,12)(8,13,21,28,22,29,24,31)(18,33,34,25,35,32,36,27);(2,4)(5,7,10,19,11,20,9,23)(6,12,16,17,15,26,14,30)(8,18,21,36,22,35,24,34)(13,27,28,32,29,25,31,33);(5,8)(6,13)(7,18,20,35)(9,21)(10,24)(11,22)(12,25,26,27)(14,28)(15,29)(16,31)(17,32,30,33)(19,34,23,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31239);