Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17,19,33)(8,10)(9,21)(13,27,28,36)(14,16)(18,34,24,22)(20,23)(25,29,32,35)(26,31);(1,2)(3,4)(5,6,8,12,11,14,10,16)(7,13,18,25,19,28,24,32)(9,15,20,26,23,30,21,31)(17,29,22,27,33,35,34,36);(2,4)(5,7,8,18,11,19,10,24)(6,13,12,25,14,28,16,32)(9,22,20,33,23,34,21,17)(15,27,26,35,30,36,31,29);(6,14)(12,16)(13,28)(15,30)(25,32)(26,31)(27,36)(29,35)Polycyclic presentation:
2^9|a^2=eh,b^2=e,c^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231475, #72231488, #72231495, … |
| maximal normal subgroup of | #72231475, #72231488, #72231495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406404);