Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,18)(9,14)(11,19)(12,17)(13,23)(15,27)(16,25)(20,29)(21,34)(22,32)(24,30)(26,33)(28,31)(35,36);(5,7,10,6,11,14,26,16,12,15,27,17,25,33,9,19)(8,20,23,34,24,35,36,30,21,13,29,18,28,32,22,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,22)(11,28)(12,21)(14,30)(15,34)(16,35)(17,20)(19,32)(24,25)(26,29)(27,36)(31,33);(8,21)(13,20)(18,34)(22,36)(23,29)(24,28)(30,31)(32,35)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,g^b=gi,d^c=di,e^c=eh,f^c=fhi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320223, #72320357, #72320481, … |
| maximal normal subgroup of | #72320223, #72320357, #72320481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406534);