Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233469 |
| 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×#3207 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1424 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)(6,11)(8,9)(12,15)(13,14)(16,17)(19,20)(21,22)(23,31)(24,33)(25,35)(26,29)(27,32)(28,34)(30,36);(2,4)(5,6)(7,15)(8,14)(9,11)(10,13)(12,17)(16,19)(18,20)(21,23)(22,27)(24,29)(25,33)(26,28)(30,34)(31,35)(32,36);(5,7,10,18)(6,12,13,19)(8,17,9,16)(11,15,14,20)(21,24,26,27)(22,28,29,23)(25,31,36,34)(30,32,35,33);(6,13)(11,14)(12,19)(15,20)(22,29)(23,28)(30,35)(32,33)Polycyclic presentation:
2^9|c^2=h,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=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, 140667);