Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501868 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,10)(7,13)(8,17)(11,19)(12,14)(15,20)(16,18);(1,2)(3,4)(5,6,8,18,9,11,17,12)(7,14,13,19,15,16,20,10);(2,4)(7,15)(10,19)(13,20)(14,16);(5,6)(7,16)(8,18)(9,11)(10,13)(12,17)(14,15)(19,20)Polycyclic presentation:
2^9|b^2=fh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,e^c=ei,e^d=eh,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72250011, #72252778, #72252782, … |
| maximal normal subgroup of | #72250011, #72252778, #72252782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409064);