Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499787 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #32888 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,16)(8,12)(9,19)(10,14)(11,15)(13,17)(18,20);(1,2)(3,4)(5,7,11,17)(6,12,15,19)(8,18,9,10)(13,14,16,20);(2,4)(5,8,11,9)(6,12,15,19)(7,10,17,18)(13,14,16,20);(5,9,11,8)(6,13,15,16)(7,10,17,18)(12,14,19,20)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,e^c=eh,f^d=fhi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72232683, #72234768, #72279160, … |
| maximal normal subgroup of | #72232683, #72234768, #72279160, … |
| maximal quotient of | #72232570, #72237595, #72252027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406990);