Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #269339 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,8,9)(7,10,12,11)(13,14)(15,17)(16,18)(19,20);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,15,19,16)(14,17,20,18);(6,9)(7,11)(10,12)(13,16,18,14,19,15,17,20);(1,3,4,2)(13,17,19,18)(14,15,20,16)Polycyclic presentation:
2^9|a^2=hi,b^2=ei,c^2=fi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 205708);