Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233521 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16)(21,22,27,29)(23,32,24,35)(25,31,26,30)(28,34,36,33);(2,4)(5,7,10,17)(6,11,14,19)(8,18,9,16)(12,20,13,15)(21,23,27,24)(22,28,29,36)(25,34,26,33)(30,35,31,32);(7,16)(8,9)(11,15)(12,13)(17,18)(19,20)(21,24,27,23)(22,28,29,36)(25,33,26,34)(30,35,31,32);(7,17)(11,19)(15,20)(16,18)(22,29)(28,36)(30,31)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140729);