Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10,14,18)(8,15)(9,12)(11,16)(13,17,20,19);(5,7)(6,10,12,18)(8,13)(9,14)(11,19,15,17)(16,20);(5,8)(6,11,12,15)(7,13)(9,16)(10,17,18,19)(14,20);(2,4)(10,18)(11,15)(13,20)(17,19)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=g,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72225672, #72225678, #72225679, … |
| maximal normal subgroup of | #72225672, #72225678, #72225679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386869);