Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,18,12,19,8,15)(7,20,21,35,25,34,24,30)(9,17,28,31,26,13,10,16)(14,29,32,23,33,22,36,27);(2,4)(6,13,19,17)(8,11)(9,10)(14,30,33,35)(15,16,18,31)(20,32,34,36)(21,24)(22,23)(26,28)(27,29);(5,7)(6,14)(8,21)(9,27)(10,29)(11,24)(12,25)(13,30)(15,32)(16,34)(17,35)(18,36)(19,33)(20,31)(22,28)(23,26);(5,8,12,11)(6,15,19,18)(7,21,25,24)(9,10,26,28)(13,31,17,16)(14,32,33,36)(20,35,34,30)(22,27,29,23)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231488, #72231547, #72231551, … |
| maximal normal subgroup of | #72231488, #72231547, #72231551, … |
| maximal quotient of | #72250015, C2×#494342, #72309695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401538);