Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,11,12,10)(8,9)(13,14)(15,19)(16,18)(17,20);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17,18,19)(16,20);(2,4)(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72236724, #72236742, #72236764, … |
| maximal normal subgroup of | #72236724, #72236742, #72236764, … |
| maximal quotient of | #72237637, #72250633, #72250636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13739);