Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,15,11,16,10,14)(7,12,18,29,19,28,20,27)(8,21,24,31,25,32,23,13)(17,33,35,30,22,36,34,26);(1,2)(3,4)(5,7,10,20,11,19,9,18)(6,12,14,27,16,28,15,29)(8,22,24,34,25,17,23,35)(13,30,21,36,31,26,32,33);(1,3,4,2)(5,8)(6,13)(7,17)(9,24)(10,23)(11,25)(12,26)(14,32)(15,21)(16,31)(18,35)(19,22)(20,34)(27,36)(28,30)(29,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30939);