Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495197 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,12,6,24)(3,5,10,22,16,23,15,13)(4,17,21,25,18,31,20,9)(14,28,26,32,29,27,30,19);(1,3,13,2)(4,14,27,9)(5,11,6,15)(7,10,23,24)(8,16,22,12)(17,20,30,19)(18,29,28,25)(21,26,32,31);(1,4,6,20,8,18,7,21)(2,9,11,17,12,25,24,31)(3,14,15,30,16,29,10,26)(5,19,22,28,23,32,13,27);(4,18)(9,25)(14,29)(17,31)(19,32)(20,21)(26,30)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72236890, #72237542, #72237552, … |
| maximal normal subgroup of | #72236890, #72237542, #72237552, … |
| maximal quotient of | #72312568, #72312687, C2×#495204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402400);