Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233501 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1286 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)(7,15)(8,18)(9,16)(10,13)(11,12)(14,20)(17,19)(21,22)(23,32)(24,33)(25,30)(26,36)(27,31)(28,34)(29,35);(2,4)(5,7,10,17)(6,11,13,8)(9,18,20,12)(14,19,16,15)(21,23,27,29)(22,28,31,24)(25,34,26,33)(30,35,36,32);(5,8,10,11)(6,12,13,18)(7,16,17,14)(9,15,20,19)(21,24,27,28)(22,29,31,23)(25,35,26,32)(30,34,36,33);(6,13)(7,17)(12,18)(14,16)(21,25,27,26)(22,30,31,36)(23,33,29,34)(24,35,28,32)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,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, 140715);