Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17)(7,19,16,9)(8,23)(10,15,12,22)(13,21,29,18)(14,30)(20,28)(24,27)(25,36)(26,34)(31,32)(33,35);(2,4)(5,7,12,21,13,22,9,6,11,16,10,18,29,15,19,17)(8,24,26,34,27,23,35,36,28,32,30,14,31,20,25,33);(5,8,10,26,13,27,19,35,11,28,12,30,29,31,9,25)(6,14,16,23,18,33,15,32,17,34,7,20,21,36,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321477, #72321498, #72321511, … |
| maximal normal subgroup of | #72321477, #72321498, #72321511, … |
| maximal quotient of | #72210531, #72210543, #72210551 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54299);