Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #299067 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #25720 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,10,11)(7,12,8,9)(14,20)(15,16)(17,18);(1,2)(3,4)(5,7,9,6,10,8,12,11)(13,14,17,20,19,16,18,15);(2,4)(6,11)(7,8)(13,15,18,16,19,20,17,14);(5,8,9,11,10,7,12,6)(13,16,18,20,19,14,17,15)Polycyclic presentation:
2^9|a^2=hi,b^2=e,c^2=fhi,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,d^c=dh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 206380);