Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,7)(8,9)(10,19)(12,15)(13,16)(14,18)(17,20);(7,15)(8,17)(9,20)(13,18);(2,4)(5,6,10,16,11,12,19,14)(7,8,18,9,15,17,13,20);(5,7)(6,8)(9,16)(10,18)(11,15)(12,17)(13,19)(14,20)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,d^b=di,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,e^d=ei,f^d=fh,g^d=ghi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224588, #72250058, #72250074, … |
| maximal normal subgroup of | #72224588, #72250058, #72250074, … |
| maximal quotient of | #72224579 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419310);