Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,28,12,18,26,17)(7,19,25,31,22,34,36,33)(8,14,20,15,9,16,10,27)(13,29,35,21,30,24,32,23);(1,2)(3,4)(5,7,20,29,11,25,9,21,12,22,10,24,26,36,8,23)(6,13,14,33,17,32,27,34,18,30,16,31,28,35,15,19);(2,4)(6,14)(7,21)(8,20)(9,10)(11,26)(13,30)(15,17)(16,18)(19,31)(22,23)(24,36)(25,29)(27,28)(33,34);(7,22)(13,30)(19,34)(21,23)(24,29)(25,36)(31,33)(32,35)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,d^b=di,e^c=eh,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250059, #72250080, #72250107, … |
| maximal normal subgroup of | #72250059, #72250080, #72250107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406510);